• 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA 

GIF^T  01^ 


Class 


APPLETOJfS'  MATHEMATICAL  SERIES. 


MENTAL   ARITHMETIC. 


BY 

G.  P.  QUACKENBOS,  LL.  D., 

AUTHOR  OF 

'•*A2f     ENGLISH    GRAMMAR',"    "FIRST    LESSONS   IN    COMPOSITION;"    "ADVANCED 
COURSE    OF    COMPOSITION    AND    RHETORIC;"    u  A    NATURAL    PHILOSO- 
PHY;"  "ILLUSTRATED  SCHOOL  HISTORY  OF  THE  UNITED 
STATES;"   " PRIMARY  HISTORY  OF  THE  UNITED 
STATES,"  ETC. 


NEW  YORK: 

D.    APPLETON    AND    COMPANY, 
549  &  551  BROADWAY. 

1878. 


[ADVERTISEMENT.] 

APPLETONS'  MATHEMATICAL  SERIES. 

BY  G.  P.  QUACKEHBOS,  LL.  D., 


A  Primary   Arithmetic.  Beautifully  illustrated;    carries 

the  beginner  through  the  first  four  Rules  and  the  simple 

Tables,  combining  mental  exercises  with  examples   for  the 

slate.     16mo.     108  pages.  22  cents. 

An  Elementary  Arithmetic.  Reviews  the  subjects  of  the 
Primary  in  a  style  adapted  to  somewhat  maturer  minds.  Also 
embraces  Fractions,  Federal  Money,  Reduction,  and  the  Com- 
pound Rules.  12mo.  144  pages.  40  cents. 

A  Practical  Arithmetic.  Prepared  with  direct  reference 
to  the  wants  of  Common  Schools,  giving  special  prominence 
to  the  branches  of  Mercantile  Arithmetic.  12mo.  336 
pages.  80  cents 

A  Mental  Arithmetic.  Designed  to  impart  readiness  in 
mental  calculations,  and  extending  them  to  the  various  opera- 
tions needed  in  business  life.  Introduces  short  methods,  and 
new  and  beautiful  processes.  16mo.  168  pages.  35  cents. 

A  Higher  Arithmetic.    I2mo.    420  pages.    $1.10. 


Entered,  according  to  Act  of  Congress,  in  the  year  1868,  by 

D.  APPLETON  &  CO., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the 
Southern  District  of  New  York. 


PREFACE. 


THE  study  of  Mental  Arithmetic  has  two  principal  objects  in 
view,  to  discipline  the  mind  and  impart  quickness  and  accuracy  in 
mental  calculations.  To  secure  these  objects  in  the  highest  degree 
and  in  the  pleasantest  way  to  both  teacher  and  learner  is  the  aim  of 
this  little  volume.  It  is  intended  for  pupils  who  have  gone  through 
a  Primary  Arithmetic,  and  know  how  to  read  and  write  numbers  up 
to  thousands  inclusive ;  and  may  be  used  either  by  itself  to  succeed 
the  Primary  or  Elementary,  or  as  an  auxiliary  to  the  Elementary  or 
Practical,  on  alternate  days. 

Among  the  more  important  features  of  the  present  work  are  the 
following: — 1.  The  gradual  and  inductive  mode  of  unfolding  the 
subject,  with  the  aid  of  rigid  but  clear  analyses  expressed  as  briefly 
as  possible.  2.  The  introduction  of  necessary  definitions,  too  often 
excluded  from  Mental  Arithmetics.  3.  The  great  variety  and  prac- 
tical bearing  of  the  Examples  ;  the  careful  avoidance  of  obscurity 
in  wording  them,  and  the  exclusion  of  all  questions  involving  im- 
possibilities or  absurdities.  4.  The  presentation  of  the  Metric  Sys- 
tem, hitherto  confined  mainly  to  text-books  on  written  Arithmetic. 
5.  The  teaching  of  short  methods  and  processes  actually  used  in  the 
counting-room.  6.  The  extension  of  mental  calculations  to  more  of 
the  operations  of  every-day  business  life  than  has  hitherto  been  at- 
tempted; such  as  equation  of  payments,  stockjobbing,  U.  S.  securi- 
ties, taxes,  duties,  &c.  The  value  of  this  last  feature,  it  is  believed, 
can  hardly  fail  to  be  appreciated  in  this  practical  age. 

The  interest  and  profit  with  which  classes  will  use  this  work  will 
depend  entirely  on  the  thoroughness  with  which  the  successive  steps 
are  taken.  Review  again  and  again  if  necessary,  and  let  nothing 
pass  till  it  is  mastered.  Short  lessons  should  be  given,  to  be  pre- 
pared beforehand.  The  books  should  be  closed  at  recitation,  the 

153.& 


4  CONTENTS. 

question  read  but  once,  and  the  scholars  should  have  no  intimation 
as  to  which  of  their  number  will  be  called  on  to  solve  it.  The  analy- 
ses given  in  the  Models  should  be  followed  (unless  better  ones  can 
be  devised),  with  distinct  articulation  and  in  correct  language.  Let 
the  answer  always  be  distinctly  stated,  when  it  is  reached,  in  con- 
nection with  the  denomination,  as  in  the  Model.  Whenever  any 
particular  form  of  analysis  has  become  perfectly  familiar,  it  will  be 
well  to  omit  it  in  the  case  of  some  of  the  questions,  and  require  im- 
mediate answers,  as  well  to  encourage  quickness  of  thought  as  to 
economize  time.  A  few  questions  from  previou^kssons,  to  be  an- 
swered thus  promptly,  will  be  found  useful  at  each  recitation. 

The  Author  can  only  hope  that  this  work  may  meet  with  as 
cordial  a  reception  as  has  been  so  kindly  extended  to  the  other 
Numbers  of  the  Series. 

NEW  YORK,  May  22,  1868. 

.     , 


CONTENTS. 


PAGl 

CHAPTER  FIRST, 

ADDITION,     

5 

CHAPTER  SECOND, 

SUBTRACTION,    

13 

CHAPTER  THIRD, 

MULTIPLICATION,    .... 

20 

CHAPTER  FOURTH, 

DIVISION,  

28 

CHAPTER  FIFTH, 

FRACTIONS,    

38 

CHAPTER  SIXTH, 

FEDERAL  MONEY,      .... 

72 

CHAPTER  SEVENTH, 

REDUCTION,   

75 

CHAPTER  EIGHTH, 

THE  METRIC  SYSTEM, 

89 

CHAPTER  NINTH, 

THE  COMPOUND  RULES,  . 

94 

CHAPTER  TENTH, 

MISCELLANEOUS  EXAMPLES, 

100 

CHAPTER  ELEVENTH, 

PERCENTAGE,  

117 

CHAPTER  TWELFTH, 

INTEREST  .                 .... 

127 

CHAPTER  THIRTEENTH, 

DISCOUNT,      ..... 

140 

CHAPTER  FOURTEENTH, 

STOCKS,  —  U.  S.  SECURITIES, 

143 

CHAPTER  FIFTEENTH, 

MISCELLANEOUS  EXAMPLES,    . 

150 

MENTAL    AEITHMETIO. 


CHAPTER      FIRST. 

ADDITION. 

[!T  is  supposed  that  the  pupil  has  learned  the  Addition,  Sub- 
traction, Multiplication,  and  Division  Tables.  Let  them  be  re- 
viewed, as  presented  under  Chapters  1,  2,  3,  and  4,  until  he  can  say 
them  promptly  and  perfectly,  backward  as  well  as  forward.] 

ADDITION  TABLE. 


1  and 

2  and 

S  and 

4  and 

5  and 

1  are     2 

1  are     3 

1  are    4 

1  are     5 

1  are     6 

2  are     3 

2  are     4 

2  are     5 

2  are     6 

2  are    7 

3  are    4 

3  are     5 

3  are     6 

3  are    7 

3  are     8 

4  are     5 

4  are     6 

4  are     7 

4  are    8 

4  are    9 

5  are    6 

5  are     7 

5  are     8 

5  are     9 

5  are  10 

6'  are     7 

6  are     8 

6  are     9 

6  are  10 

6  are  11 

7  are    8. 

7  are     9 

7  are  10 

7  are  11 

7  are  12 

8  are     9 

8  are  10 

8  are  11 

8  are  12 

8  are  13 

9  are  10 

9  are  11 

9  are  12 

9  are  13 

9  are  14 

10  are  11 

10  are  12 

10  are  13 

10  are  14 

10  are  15 

6  and 

7  and 

8  and 

9  and 

10  and 

1  are     7 

1  are     8 

1  are     9 

1  are  10 

1  are  11 

2  are     8 

2  are     9 

2  are  10 

2  are  11 

2  are  12 

3  are     9 

3  are  10 

3  are  11 

3  are  12 

3  are  13 

4  are  10 

4  are  11 

4  are  12 

4  are  13 

4  are  14 

5  are  11 

5  are  12 

5  are  13 

5  are  14 

6  are  15 

6  are  12 

6  are  13 

6  are  14 

6  are  15 

6  are  16 

7  are  13 

7  are  14 

7  are  15 

7  are  16 

7  are  17 

8  are  14 

8  are  15 

8  are  16 

8  are  17 

8  are  18 

9  are  15 

9  are  16 

9  are  17 

9  are  18 

9  are  19 

10  are  16 

10  are  17 

10  are  18 

10  are  19 

10  are  20 

6  ADDITION. 

SECTION  1.— Addition  is  the  process  of  uniting 
two  or  more  numbers  in  one,  called  their  Sum.  Two 
and  one  are  three  /  we  have  added  2  and  1,  and  3  is 
their  sum. 

Addition  is  denoted  by  an  erect  cross  -f ,  called 
Plus,  placed  between  the  numbers  to  be  added.  2  +  1 
is  read  two  plm  one,  and  means  that  2  and  1  are  to  be 
added. 

Two  short  horizontal  lines  =  denote  equality. 
2  -f  1=3,  is  read  two  plus  one  equals  three,  and  means 
that  the  sum  of  2  and  1  is  3. 

1.  What  is  the  sum  of  2  and  3  ?     2  and  9  ? 

2.  What  is  the  sum  of  2  and  4  ?     2  and  7  ? 

3.  What  is  the  sum  of  3  and  1  ?     3  and  4  ? 

4.  What  is  the  sum  of  3  and  8  ?     3  and  6  ? 

5.  What  is  the  sum  of  4  and  5  ?     4  and  2  ? 

6.  What  is  the  sum  of  4  and  9  ?     4  and  10  ? 

7.  How  many  are  5  and  8  ?     5  and  1  ?     5  and  5  ? 

8.  How  many  are  5  and  9  ?     5  and  3  ?     5  and  6  ? 

9.  How  many  are  6  and  2  ?     6  and  9  ?     6  and  5  ? 

10.  How  many  are  6  and  1  ?     6  and  6  ?     6  and  8  ? 

11.  Add  7  and  7.     7  and  2.     7  and  4.     7  and  9. 

12.  Add  7  and  8.     7  and  1.     7  and  5.     7  and  10. 

13.  Add  8  and  8.     8  and  6.     8  and  3.     8  and  4. 

14.  Add  8  and  1.     8  and  9.     8  and  2.     8  and  7. 

15.  9  +  4=  how  many?     9  +  9?     9  +  1?     9  +  0? 

16.  9  +  7=  how  many?     9  +  2?     9  +  6?     9  +  3? 

17.  What  is  the  value  of  10  +  1?    10  +  10?    10  +  9? 
10  +  5?     5  +  10?     10  +  8?     10  +  4?     1+9?     1+5? 

18.  How  many  are  2  and  5?     5  and  2?     How 
muchis6  +  4?     4  +  6?     3  +  7?     7  +  3? 


ADDITION.  7 

19.  When  two  numbers  are  to  be  added,  does  it 
make  any  difference  which  is  taken  first  ? 

20.  How  many  are  4  and  8  ?    How  many  are  1 
and  3  and  8  ?     How  many  are  2  and  2  and  8  ? 

21.  How  many  are  3  and  9  ?     How  many  are  1 
and  2  and  9  ?     How  many  are  2  and  1  and  9  ? 

22.  How  many  are    8    and  10?      How  much   is 

3  +  5  +  10?     How  much  is  2  +  2  +  4  +  10? 

23.  How  many  are  6  and  7  ?     How  much  is  3  +  3 
+  7  ?    How  much  is  1+4  +  1  +  7? 

24.  How  many  are  9  and  8?     How  much  is  5  + 

4  +  8?     How  much  is  3  +  5  +  1+8? 

25.  How  many  are  8  and  5  ?     How  many  are  8 
and  3  +  2  ?     How  many  are  8  and  4  +  1  ? 

26.  How  many  are   10  and   7?     2  +  8  and  4  +  3? 
10andl+6?     1+6  +  10? 

27.  Mary  and  her  seven  sisters  just  filled  a  bench  ; 
how  many  did  the  bench  hold  ? 

28.  You   have  four  fingers  on  each  hand ;    how 
many  fingers  have  you  on  both  hands  ? 

29.  A  newsboy  bought  five  daily  and  seven  week- 
ly papers ;  how  many  papers  had  he  in  all  ? 

30.  If  Ruth  has  seven  pins  in  one  pin-cushion,  and 
six  in  another,  how  many  pins  has  she  in  both  ? 

31.  Guy  pulled  six  ears  of  corn  on  Monday,  and 
ten  on  Tuesday  ;  how  many  did  he  pull  both  days  ? 

32.  Ten  birds  were  sitting  on  a  steeple,  when  six 
more  alighted ;  how  many  were  there  then  ? 

33.  Henry  had  two  cents,  and  his  father  gave  him 
ten  more.     Louis  had  ten  cents,  and  he  found  two 
more  in  the  street.     Which  then  had  the  most  ? 


ADDITIC 


34.  Put  four  marbles  in  a  bag  already  containing 
ten,  and  how  many  will  you  have  in  the  bag  ? 

35.  A  boy  caught  six  perch,  three  pickerel,  and  ten 
shiners ;  how  many  fish  did  he  catch  in  all  ? 

36.  A  baker  gave  nine  loaves  to  one  poor  family, 
and  five  to  another ;  how  many  did  he  give  both  ? 

37.  How  many  trees  stand  beside  my  lane,  if  there 
are  ten  on  one  side  and  three  on  the  other  ? 

38.  A  mother,  having  three  sons  aged  two,  five, 
and  eight  years,  gave  each  as  many  dollars  as  he  was 
years  old  ;  how  many  dollars  did  she  give  all  three  ? 

39.  How  many  books  had  Paul,  if  his  father  gave 
him  3,  his  mother  6,  his  brother  1,  and  his  sister  7  ? 

40.  A  farmer  had  four  ducks,   six  geese,  and  as 
many  chickens  as   he  had  ducks  and  geese  put  to- 
gether ;  how  many  fowls  had  he  in  all  ? 


SECTION  2.— 1.  How  many  are  3  and  2  ?     13  and 
2  ?     33  and  2  ?     63  and  2  ?     93  and  2  ?     103  and  2  ? 

2.  What  does  2  +  3  equal?     22  +  3?     43  +  2?     53 
+  2?     83+2?     183  +  2?     193  +  2?     203  +  2? 

3.  What  is  the  sum  of  4  and  5  ?     5  and  4?     55 
and  4  ?     54  and  5  ?     74  and  5  ?     75  and  4  ? 

4.  How  much  is  7  +  2?     107  +  2?     207  +  2?     507 
+  2?     807  +  2?     1007  +  2?     4007  +  2?     5007  +  2? 

5.  Add  2  and  6.     62  and  6.     162  and  6.     1062  and 
6.     1066  and  2.     166  and  2.     66  and  2.     76  and  2. 

6.  How  much  is  3  +  4?      20  +  3  +  4?      30  +  3  +  4? 
40  +  3  +  4?     3  +  40  +  4?     60  +  4  +  3?     4  +  3  +  70? 

7.  How  much  is  5  +  2?     50  +  5+2?     51+5  +  2? 


ADDITION.  y 

8.  How  many  are  9  and  1  ?     89  and  1  ?     99  and  1  ? 
109  and  1  ?     199  and  1  ?     209  and  1  ?     299  and  1  ? 

9.  How  many  are  6  and  4  ?     6  and  5  ?     16  and  4  ? 
16  and  5  ?     26  and  5  ?     36  and  5  ?     56  and  5  ? 

10.  How  many  are  8  and  7  ?     18  and  7  ?     118  and 
7  ?     128  and  7  ?     28  and  7  ?     38  and  7  ?     638  and  7  ? 

11.  What  is  3  +  9  equal  to?     4  +  9?     23  +  9?     24 
+  9?     54  +  9?     55  +  9?     56  +  9?     52  +  9? 

12.  What  is  2  +  5  +  7  equal  to?     12  +  5  +  7?     22  + 
5  +  7?     42  +  5  +  7?     72  +  5  +  7?     92  +  5  +  7?  . 

13.  How  many  are  63  and  5  ?     79  and  2  ?    103  and 
8?     47  and  6?     99  and  3  ?     102  and  7?     113  and  5? 

14.  How  many  are  10  and  10?     20  and  10?     50 
and  10?     90  and  10?     100  and  10  ?     200  and  10  ? 

15.  6  and  9  make  how  many  ?     34  and  6  ?     43  and 

9  ?     52  and  8  ?     58  and  2  ?     50  and  3  ?     79  and  6  ? 
88  and  4  ?     103  and  3  ?     Ill  and  9  ?     124  and  4  ? 

16.  How  many  are  7  and  7  and  8?     21  and  9  and 
6?     34  and  8  and  7  ?     41  and  10  and  9?     72  and  4 
and  5  ?     99  and  10  and  1  ?     199  and  2  and  9  ? 

17.  One  tree  bears  34  apples,  another  6,  and  a  third 

10  ;  how  many  apples  do  all  three  bear? 

MODEL.  If  one  tree  bears  34  apples,  another  6,  and  a  third  10, 
all  three  together  will  bear  the  sum  of  34,  6,  and  10  apples,  or  50 
apples.  Amwer,  50  apples. 

18.  A  farmer  set  out  42  apple,  8  pear,  and  9  plum 
trees ;  how  many  trees  did  he  set  out  altogether  ? 

19.  If  I  travel  75  miles  by  boat,  7  by  railroad,  and 
10  by  stage,  how  many  miles  do  I  go  in  all  ? 

20.  103  English  books,  20  French  books,  and  6 
German  books,  make  how  many  books  in  all  ? 


10  ADDITION. 

21.  James  bought  20  cents'  worth  of  bread,   10 
cents'  worth  of  cake,  and  9  cents'  worth  of  crackers ; 
how  much  did  he  have  to  pay  the  baker  ? 

22.  How  many  miles  will  you  travel  in  a  day,  if 
you  go  21  miles  in  the  morning,  30  miles  in  the  after- 
noon, and  8  miles  in  the  evening  ? 

23.  In  a  field  were  19  sheep,  8  cows,  and  5  calves  ; 
how  many  animals  were  in  the  field  ? 

24.  John  was  eight  years  older  than  Charles.    How 
old  was  John  when  Charles  was  thirteen  ?     How  old 
was  each,  nine  years  afterward  ? 

25.  How  many  windows  in  a  factory,  if  there  are 
12  in  front  and  6  on  each  of  the  other  three  sides  ? 

26.  In  a  certain  school  were  10  boys  and  6  more 
girls  than  boys ;  how  many  scholars  altogether  ? 

27.  Helen  laid  out  45  cents  for  muslin,  5  cents  for 
thread,  9  cents  for  pins,  and  6  cents  for  needles ;  how 
many  cents  did  she  spend  in  all  ? 

28.  A  fisherman  had  sold  all  his  stock,  except  17 
shad,  6  bass,  and  9  eels ;  how  many  fish  had  he  left  ? 

29.  Count  by  2's,  beginning  2,  4,  6,  &c.,  up  to  100. 

30.  Count  by  2's,  beginning  1,  3,  5,  &c.,  up  to  99. 

31.  Count  by  3's,  beginning  1,  4,  7,  &c.,  up  to  100. 

32.  Count  by  4's,  beginning  2,  6,  10,  &c.,  up  to  98. 

33.  Count  by  5's,  beginning  1,  6,  11,  &c.,  up  to  96. 

34.  Count  by  6's,  beginning  2,  8, 14,  &c.,  up  to  98. 

35.  Count  by  7's,  beginning  3, 10,  17,  &c.,up  to  94. 

36.  Count  by  8's,  beginning  5,  13,  21,  &c.,  up  to  93. 

37.  Count  by  9's,  beginning  1, 10, 19,  &c.,  up  to  100. 
[These  exercises  may  be  continued,  and  varied  by  commencing 

differently,  till  they  are  made  perfectly  familiar.] 


ADDITION.  1 1 

SECTION  3. — 1.  How  many  are  43  and  26  ? 

MODEL.  43=3  units  4  tens;  26=6  units  2  tens.  6  units  and 
3  units  are  9  units ;  2  tens  and  4  tens  are  6  tens.  6  tens  9  units 
are  69.  Ans.  69. 

2.  What  is  the  sum  of  31  and  52  ?     Of  74  and  1 5  ? 

3.  How  many  are  22  and  60?    17  +  81?     65  +  32? 

4.  How  many  are  14  and  54?     46  +  31?     80  +  19? 

5.  Add  123  and  876.     456  and  543.     739  and  40. 

6.  What  is  the  sum  of  267  and  431  ? 

SHORT  FORM.  1  and  7  are  8 ;  3  and  6  are  9 ;  4  and  2  are  6. 
Ans.  698. 

7.  How  many  are  240  and  326  ?     579  and  120  ? 

8.  How  many  are  165  and  722  ?     410  and  378  ? 

9.  How  many  are  821  and  73  ?     26  and  933  ? 

10.  What  is  the  sum  of  59  and  63  ? 

SHORT  FORM.  3  and  9  are  12,  2  units  and  1  ten ;  1  and  6  and  5 
are  12.  Ans.  122. 

11.  How  many  are  14  and  18?     17  +  23?     21  +  16? 

12.  How  many  are  40  and  69?     53  +  29?    37  +  47? 

13.  How  many  are  99  and  12?     86  +  62?    29  +  59? 

14.  How  many  are  108  and  15?   221+49?  72  +  68? 

15.  How  many  are  29  and  29?    38  +  57?    93  +  67? 

16.  Count  by  tens, — 10,  20,  30,  &c., — up  to  100. 

17.  Count  by  elevens, — 11,  22,  33,  &c.5 — up  to  132. 

18.  Count  by  twelves,— 12,  24,36,  <fcc.,— up  to  144. 

19.  Andrew  had  23  marbles,  and  won  38  more; 
how  many  had  he  then  ? 

20.  If  I  give  57  cents  to  one  poor  family,  and  39 
cents  to  another,  how  much  do  I  give  both  ? 

21.  Which  is  greater,  26  +  57  or  77  +  16  ? 

22.  After  spending  18  dollars  for  groceries,  and  27 


1 2  ADDITION. 

dollars  for  drygoods,  a  lady  found  that  she  had  five 
dollars  left.     How  many  dollars  had  she  at  first  ? 

23.  How  many  birds  are  in  two  flocks,  one  con- 
taining 97  birds,  and  the  other  64  ? 

24.  A  drover  who  had  56  sheep,  bought  14  more, 
and  another  flock  of  30 ;  how  many  sheep  had  he  then  ? 

25.  Noah  lived  600  years  before  the  Flood,  and 
350  years  after  it ;  how  old  was  Noah  when  he  died  ? 

26.  Methuselah  lived  19  years  more  than  Noah; 
what  age  did  Methuselah  attain  ? 

27.  A  man  who  gave  125  dollars  for  a  watch,  sold 
it  for  25  dollars  more  than  it  cost ;  what  did  he  get 
for  it  ? 

28.  How  many  pounds  are  there  in  two  bales  of 
cotton,  one  weighing  404  pounds,  and  the  other  382  ? 

29.  If  a  farmer  raises  93  bushels  of  wheat  and  128 
of  corn,  how  many  bushels  of  both  does  he  raise? 

30.  A  lot  was  bought  for  360  dollars,  and  sold  at 
a  profit  of  75  dollars  ;  how  much  did  it  bring? 

MODEL.  If  it  was  bought  for  360  dollars  and  sold  at  a  profit  of 
75  dollars,  it  must  have  brought  the  sum  of  360  and  75  dollars,  or 
435  dollars.  Am.  435  dollars. 

31.  A  horse  was  bought  for  238  dollars,  and  sold  at 
a  profit  of  27  dollars ;  what  was  it  sold  for? 

32.  A  man  makes  56  dollars  by  selling  some  goods 
that  cost  him  249  dollars ;  what  does  he  sell  them  for? 

33.  An  orchard  cost  275  dollars;  the  profit  on  it 
being  87  dollars,  what  was  it  sold  for  ? 

34.  Conrad  had  8  books,  containing  63  pictures, 
and  19  books,  containing  148  pictures.     How  many 
books  had  Conrad  in  all,  and  how  many  pictures? 


SUBTRACTION. 


13 


CHAPTER     SECOND.. 

STJBTBACTION. 

SECTION  4.— Subtraction  is  the  process  of  taking 
one  number  from  another. 

Subtraction  is  denoted  by  a  short  horizontal 
line  —  ,  called  Minus,  placed  before  the  smaller 
number. 

4—1  is  read  four  minus  one,  and  means  that  1  is 
to  be  subtracted  from  4. 

4—1=3.  We  have  subtracted  1  from  4;  the  re- 
sult, 3,  is  called  the  Remainder  or  Difference. 

SUBTRACTION  TABLE. 


1  from 

2  from 

3  from 

4  from 

5  from 

1  leaves  0 

2  leaves  0 

3  leaves  0 

4  leaves  0 

5  leaves  0 

2  leaves  1 

3  leaves  1 

4  leaves  1 

5  leaves  1 

6  leaves  1 

3  leaves  2 

4  leaves  2 

5  leaves  2 

6  leaves  2 

7  leaves  2 

4  leaves  3 

6  leaves  3 

6  leaves  3 

7  leaves  3 

8  leaves  3 

6  leaves  4 

6  leaves  4 

7  leaves  4 

8  leaves  4 

9  leaves  4 

6  leaves  5 

7  leaves  5 

8  leaves  5 

9  leaves  5 

10  leaves  5 

7  leaves  6 

8  -leaves  6 

9  leaves  6 

10  leaves  6 

11  leaves  6 

8  leaves  7 

9  leaves  7 

10  leaves  7 

11  leaves  7 

12  leaves  7 

9  leaves  8 

10  leaves  8 

1  1  leaves  8 

12  leaves  8 

13  leaves  8 

10  leaves  9 

11  leaves  9 

12  leaves  9 

13  leaves  9 

14  leaves  9 

6  from 

7  from 

8  from 

9  from 

10  from 

6  leaves  0 

7  leaves  0 

8  leaves  0 

9  leaves  0 

10  leaves  0 

7  leaves  1 

8  leaves  1 

9  leaves  1 

10  leaves  1 

11  leaves  1 

8  leaves  2 

9  leaves  2 

10  leaves  2 

11  leaves  2 

12  leaves  2 

9  leaves  3 

10  leaves  3 

11  leaves  3 

12  leaves  3 

13  leaves  3 

10  leaves  4 

11  leaves  4 

12  leaves  4 

13  leaves  4 

14  leaves  4 

11  leaves  5 

12  leaves  5 

13  leaves  5 

14  leaves  5 

15  leaves  5 

12  leaves  6 

13  leaves  6 

14  leaves  6 

15  leaves  6 

16  leaves  6 

13  leaves  7 

14  leaves  7 

15  leaves  7 

16  leaves  7 

17  leaves  7 

14  leaves  8 

15  leaves  8 

16  leaves  8 

17  leaves  8 

18  leaves  8 

15  leaves  9 

16  leaves  9 

17  leaves  9 

18  leaves  9 

19  leaves  9 

14  SUBTRACTION. 

1.  How  many  does  3  from  5  leave?     4  from  8? 
7  from  9  ?     1  from  4  ?     2  from  6  ?     5  from  7  ? 

2.  Take  I  from  9.     2  from  10.    3  from  11.    9  from 
18.     1  from  14.     10  from  15.     6  from  15.     4  from  13. 

3.  How  much  is  7  less  4  ?     17  less  4  ?     27  less  4  ? 
47  less  4  ?     87  less  4  ?     37  less  4  ?     97  less  4  ? 

4.  How  much  is  8-2?    98-2?    108-2?    68-2? 
168  — 2?     268  —  2?     768  —  2?     778  —  2? 

5.  Subtract  7  from  9.     27  from  29.     97  from  99. 
~107  from  109.     7  from  8.     207  from  208. 

6.  How  much  is  11 -10?     31-10?     51-10?     81 
— 10?       13  —  10?       23  —  10?       123  —  10?      223  —  10? 

7.  How  much  remains,  if  we   take  20  from  40  ? 
:.20  from  47  ?    20  from  67  ?    20  from  89  ?    20  from  39  ? 

8.  30  from  50  leaves   how  many?     50  from  90? 
;^0  from  91  ?     60  from  82  ?    40  from  78  ?    70  from  83  ? 

#.  21  from  43?  56  from  67?  71  from  85?  38 
lifrom  59  ?  86  from  98  ?  31  from  77  ?  44  from  59  ? 

10.  If  a  man  bought  a  lamp  for  5  dollars  and  sold 
Sit  for  8,  how  many  dollars  did  he  make  ? 

MODEL.  If  a  man  bought  a  lamp  for  5  dollars  and  sold  it  for  8, 
1  he  must  have  made  the  difference  between  5  and  8  dollars,  or  3 
•  dollars.  Am.  3  dollars. 

11.  What  is  the  profit  on  a  barrel  of  flour,  bought 
for  8  dollars  and  sold  for  12  dollars? 

12.  Mary  is  13  years  old  and  Sarah  19 ;  two  years 
I  hence,  what  will  be  the  difference  in  their  ages? 

13.  How  many  more  boys  than  girls  are  there  in  a 
;  school,  if  there  are  75  boys  and  32  girls? 

14.  In  a  stable  there  were  23  horses  and  46  stalls; 
iJhow  many  empty  stalls  were  there  ? 


SUBTRACTION.  15 

15.  A  gentleman  now  87  years  old  was  married 
56  years  ago  ;  how  old  was  he  when  he  was  married? 

16.  A  boy  carrying  39  eggs  in  a  basket,  fell  and 
broke  19  of  them ;  how  many  were  left  whole  ? 

1 7.  Ninety-three  persons  were  wrecked  in  a  storm. 
Eighty  were  saved ;  how  many  perished  ? 

18.  A  person  who  had  88  melons,  sold  67  of  them ; 
how  many  had  he  left  ? 

19.  How  much  does  a  merchant  make  on  goods 
bought  for  68  dollars  and  sold  for  88? 

20.  A  drover,  having  96  sheep,  divided  them  into 
2  flocks.     He  put  25  in  one  flock ;  how  many  in  the 
other  ?     If  from  the  second  flock  he  sold  40,  how  many 
had  he  left  in  that  flock  ?     How  many  in  all  ? 

21.  What  change  must  a  boy,  who  pays  15  cents 
for  a  ball,  get  for  a  25-cent  stamp  ? 


SECTION  5.— 1.  How  much  does  3  from  12  leave? 
3  from  22  ?     3  from  32  ?     3  from  52  ?     3  from  72  ? 

2.  5  from  11  leaves  how  many?     5  from  21?     5 
from  51  ?     5  from  91  ?     5  from  31  ?     5  from  71  ? 

3.  7  from  10?     7  from  20?     7  from  80?     7  from 
60  ?    7  from  30  ?    7  from  70  ?    7  from  50  ?    7  from  40  ? 

4.  8  from  14?     8  from  54?     8  from  94?     8  from 
24  ?    2  from  11  ?    2  from  71  ?    2  from  31  ?    2  from  81  ? 

5.  How  much  is  10—9?     30—9?     60—9?     13— 
4?     93—4?     83  —  4?     83  —  5?     13  —  5? 

6.  How  much  is  12  — 6?    42  —  6?    62  —  6?    17—9? 
57  —  9?    37  —  9?    10  —  2?    90  —  2?    100—2?    200—2? 
16  —  8?    100—8?    300—8?    12  —  7?    92  —  7?    102  —  7? 


16  SUBTRACTION. 

7.  Take  5   from  10.     5  from  .30.     5  from  130.     5 
from  230.     6  from  14.     6  from  114.     6  from  214. 

8.  How  much  is  11  less  7?     Ill  less  7?    411  less 

7  ?     15  less  9  ?     115  less  9  ?     215  less  9  ?     25  less  9  ? 

9.  How  much  does  8  from  12  leave?     8  from  42? 

8  from  142  ?     3  from  11  ?     3  from  81  ?     3  from  101  ? 

10.  34  from  41  leaves  how  many?     24  from  51  ? 
64  from  91  ?     14  from  71  ?     54  from  81  ? 

11.  15  from   32?     35  from  72?     55  from  92  ?     26 
from  33  ?     46  from  63  ?     16  from  93  ?     86  from  103  ? 

12.  37  from  94  ?     67  from  84  ?     18  from  45  ?     28 
from  85?     4  from  10?     4  from  100?     24  from  70  ? 

13.  How  much  is  17— 8?     47—18?     67—28?     87 
—38?     42—5?     42  —  25?     72—55?     92  —  35? 

14.  Take  14  from  42.    114  from  142.    34  from  142. 
44  from  152.     25  from  34.     65  from  84. 

15.  9  from   13  leaves  how  many  ?     29  from  53  ? 
11  from  60  ?     13  from  80  ?     41  from  90  ?  74  from  80  ? 
26  from  90  ?    17  from  36  ?    49  from  93  ?    29  from  71  ? 

16.  When  subtraction  is  to  be  denoted,  what  sign 
is  used  ?    Which  number  is  it  placed  before  ? 

17.  What  sign  is  used,  when  addition  is  to  be  de- 
noted ?    What  is  the  sign  of  equality  ? 

18.  Count  backward  by  2's  from  100:  100,  98,  &c. 

19.  Count  backward  by  3's  from  100:  100,  97,  <fcc. 

20.  Count  backward  by  4's  from  100  :  100,  96,  &c. 

21.  Count  backward  by  4's  from  99  :  99,  95,  91,  &c. 

22.  Count  backward  by  5's  from  100  :  100,  95,  &c. 

23.  Count  backward  by  6's  from  100  ;  from  95. 

24.  Count  backward  by  7's  from  100 ;  from  98. 

25.  Count  backward  by  8's  from  100  ;  by  9's. 


SUBTRACTION.  17 

26.  A  man  bought  a  cow  for  $75,*  and  paid  $29 
on  account.     How  much  remained  due  ? 

MODEL.  If  a  man  bought  a  cow  for  $75  and  paid  $29  on  ac- 
count, there  remained  due  the  difference  between  $75  and  $29,  or 
$46.  Am.  $46. 

27.  From  New  York  to  Troy  by  the  Hudson  River 
E-ailroad  is  150  miles;  from  New  York  to  Poughkeep- 
sie  by  the  same  road  is  74  miles.     How  far  is  it  from 
Poughkeepsie  to  Troy  ? 

28.  A  owes  B  $47,  and  pays  him  on  account  $18  ; 
how  much  does  he  then  owe  B  ? 

29.  If  63  melons  were  stolen  from  a  pile  contain- 
ing 100,  how  many  were  left  ? 

30.  If  you  have  used  88  pens  out  of  a  box  con- 
taining 144,  how  many  are  left  ? 

31.  A  pole  72  inches  long  is  driven  into  the  ground 
24  inches;  how  many  inches  of  the  pole  are  left  above 
ground?    If  14  inches  are  then  cut  off,  how  many 
will  be  left  above  ground  ? 

32.  C  owes  D  $78.     If  he  hands  D  a  hundred-dol- 
lar bill,  how  much  change  should  he  receive? 

33.  If  17  pounds  of  butter  are  used  out  of  a  firkin 
that  held  63  pounds,  how  many  pounds  are  left  ? 

34.  Charles  is  25  years  younger  than  his  father;, 
how  old  is  he,  when  his  father  is  73  ? 

35.  A  hardware  merchant  sold  for  $101  some  iron 
that  cost  $88 ;  what  was  his  profit  ? 

36.  What  number  must  I  add  to  111,  to  produce 
132?     To  produce  144? 

*  This  mark  ($)  denotes  dollars.  It  is  always  placed  before  the  number. . 
$75  is  read  seventy-Jive  dollars. 


18  SUBTRACTION. 

SECTION  6,— i.  How  much  is  8  +  9— 7? 

MODEL.     8  +  9  equals  17.     17—7  equals  10.     Ans.  10. 

2.  How  much  is  6  +  13  — 8?     14  +  8  —  12? 

3.  How  much  is  15  — 9  +  8?     23  —  11  +  7? 

4.  How  much  is  34  +  12— 9?     47  +  11  —  12? 

5.  How  much  is  23+45—32  ?     56  + 12—35  ? 

6.  How  much  is  31  +  17— 25?     73  +  26—52? 

7.  How  much  is  42  +  39— 28?     67  +  24—19? 

8.  How  much  is  76  +  22— 49?     18  +  56  —  36? 

9.  From  the  sum  of  47  and  58  subtract  the  sum 
of  57  and  48. 

10.  From  the  sum  of  26  and  36  subtract  the  dif- 
ference between  50  and  25. 

11.  From  the  difference  between  26  and  88  take 
the  sum  of  17  and  27. 

12.  From  the  difference  between  101  and  11  take 
the  difference  between  98  and  77. 

13.  From  the  difference  between  84  and  16  take 
the  difference  between  45  and  36. 

14.  After  a  battle,  18  men  in  a  certain  company 
were  found  to  be  killed,  and  34  wounded.     If  the 
company  contained  89  men,  how  many  were  uninjured  ? 

MODEL.  If  18  men  were  killed,  and  34  wounded,  the  total  of 
killed  and  wounded  was  18  +  34,  or  52,  men;  and  the  number  un- 
injured, was  the  difference  between  89  and  52,  or  37,  men.  Ans.  37 
men. 

15.  A  person  spent  $37  for  carpeting  and  $56  for 
furniture;  what  did  the  whole  cost  him?    If  he  sold 
the  whole  for  $100,  what  was  his  profit? 

16.  If  a  man  who  has  $54  earns  $21  more,  and 
then  spends  $43,  how  much  has  he  left  ? 


SUBTRACTION.  19 

17.  James  had  96  marbles;  he  gave  away  24,  and 
lost  36.     How  many  had  he  left  ? 

18.  A  certain  boat  has  a  crew  of  18  persons,  and 
73  passengers  on  board.     If  53  persons  get  off,  how 
many  are  left  aboard  ? 

19.  P  had  $20,  and  earned  $13  more.     Q  had  $46, 
and  spent  $19.     Which  then  had  the  most,  and  how 
much  ? 

20.  An  orchard  of  apple,  pear,  and  cherry  trees, 
contains  200  trees  in  all.     If  there  are  125  apple  trees, 
and  46  pear  trees,  how  many  cherry  trees  are  there  ? 

21.  A  farmer,  who  had  made  336  pounds  of  butter, 
sold  110  pounds  to  one  customer,  and  127  to  another; 
how  many  pounds  were  left  ? 

22.  If  a  person  gets  14  quarts  of  strawberries  from 
one  bed,  19  from  another,  and  23  from  a  third,  how 
many  quarts  will  he  have  left  after  selling  37  quarts  ? 

23.  If  a  person  gets  29  quarts  of  strawberries  from 
one  bed  and  38  from  another,  how  many  quarts  will 
he  have  left  after  selling  17  quarts  and  giving  25  away  ? 

24.  A  and  B  start  from  two  places  96  miles  apart, 
and  travel  toward  each  other.     The  first  day,  A  goes 
48  miles,  and  B  27 ;  how  many  miles  are  they  then 
apart  ?    How  far  is  A  from  B's  starting-place  ?    How 
far  is  B  from  A's  starting-place  ? 

25.  What  number  must  you  add  to  the  sum  of 
nine,  thirteen,  and  forty-eight,  to  produce  99  ? 

26.  There  are  sixty-six  rows  of  corn  to  hoe  in  one 
field,  and  forty-three  in  another.     When  Robert  has 
hoed  19  rows,  Richard  23,  and  Reuben  27,  how  many 
rows  still  remain  to  be  hoed  ? 


20 


MULTIPLICATION. 


CHAPTER     THIRD. 

MULTIPLICATION. 

SECTION  7.— Multiplication  is  the  process  of  tak- 
ing a  number  a  certain  number  of  times. 

Multiplication  is  denoted  by  a  slanting  cross  x  , 
placed  between  the  numbers  to  be  multiplied  together. 

MULTIPLICATION  TABLE. 


Once 

Twice 

3  times 

4  times 

5  times 

6  times 

1  is    1 

1  is     2 

1  is     3 

1  is    4 

1  is     5 

1  is     6 

2  is     2 

2  is    4 

2  is     6 

2  is     8 

2  is  10 

2  is  12 

3  is     3 

3  is     6 

3  is     9 

3  is  12 

3  is  15 

3  is  18 

4  is    4 

4  is     8 

4  is  12 

4  is  16 

4  is  20 

4  is  24 

5  is     5 

5  is  10 

5  is  15 

5  is  20 

5  is  25 

5  is  30 

6  is     6 

6  is  12 

6  is  18 

6  is  24 

6  is  30 

6  is  36 

7  is    7 

7  is  14 

7  is  21 

7  is  28 

7  is  35 

7  is  42 

8  is    8 

8  is  16 

8  is  24 

8  is  32 

8  is  40 

8  is  48 

9  is     9 

9  is  18 

9  is  27 

9  is  36 

9  is  45 

9  is  54 

10  is  10 

10  is  20 

10  is  30 

10  is  40 

10  is  50 

10  is  60 

11  is  11 

11  is  22 

11  is  33 

11  is  44 

11  is  55 

11  is  66 

12  is  12 

12  is  24 

12  is  36 

12  is  48 

12  is  60 

12  is  72 

7  times 

8  times 

9  times 

10  times 

11  times 

12  times 

1  is    7 

1  is     8 

1  is      9 

1  is    10 

1  is    11 

lis    12 

2  is  14 

2  is  16 

2  is    18 

2  is    20 

2  is    22 

2  is    24 

3  is  21 

3  is  24 

3  is    27 

3  is    30 

3  is    33 

3  is    36 

4  is  28 

4  is  32 

4  is    36 

4  is    40 

4  is    44 

4  is    48 

5  is  35 

5  is  40 

5  is    45 

5  is    50 

5  is    55 

5  is    60 

6  is  42 

6  is  48 

6  is    54 

6  is    60 

6  is    66 

6  is    72 

7  is  49 

7  is  56 

7  is    63 

7  is    70 

7  is    77 

7  is    84 

8  is  56 

8  is  64 

Sis    72 

8  is    80 

8  is    88 

8  is    96 

9  is  63 

9  is  72 

9  is    81 

9  is    90 

9  is    99 

9  is  108 

10  is  70 

10  is  80 

10  is    90 

10  is  100 

10  is  110 

10  is  120 

11  is  77 

11  is  88 

11  is    99 

11  is  110 

11  is  121 

11  is  132 

12  is  84 

12  is  96 

12  is  108 

12  is  120 

12  is  132 

12  is  144  , 

MULTIPLICATION.  21 

2  x  3  is  read  two  multiplied  by  three,  and  means 
two  taken  3  times. 

2  x  3=6.  We  have  multiplied  2  by  3  ;  the  result, 
8,  is  called  their  Product. 

1.  How  much  is  3  times  4  ?     4  times  9  ?     Twice 
12?     4  times  6?     9  times  2  ?     8  times  4? 

2.  What  is  the  product  of  4  and  2  ?     9  and  6  ? 
6  and  7?    11  and  3?    11  and  4?    12  and  6?    8  and  3? 

3.  How  much  is  12x5?     6x8?     3x9?     8x7? 
4x10?     10x6?     5x11?     12x3?     7x7? 

4.  Multiply  8  by  8.     4  by  4.     6  by  6.     11  by  11. 
5  by  5.     3  by  3.     12  by  12.     2  by  2. 

5.  How  much  is  12  times  7  ?    7  times  12  ?    6  times 
11?     11  times  6?     3x4x2?     2x4x3? 

6.  When  numbers  are  to  be  multiplied  together, 
does  the  order  in  which  they  are  taken  affect  the 
product  ? 

7.  How  much  is  7  times  9  ?     9  times  7  ?    Twice 
6?     6  times  2?     12x4?     4x12?     3x11?     11x3? 

8.  How  much  is  10  x  9  ?    6x10?    10x7?    2x10? 
10x5?     8x10?     10x10?     10x3? 

9.  When  10  is  a  factor,  with  what  figure  does  the 
product  always  end  ? 

10.  How  much  is  5x1?  5x9?  5x3?  5x7? 
5x2?  5x6?  5x8?  6x5?  9x5? 

11.  When   5  is  a  factor,  with  one  of  what  two 
figures  does  the  product  end  ? 

12.  What  will  9  tops  cost,  at  5  cents  apiece  ? 
MODEL.     Nine  tops  will  cost  9  times  as  much  as  1  top.     If  1  top 

costs  5  cents,  9  tops  will  cost  9  times  5,  or  45,  cents.   Arts.  45  cents* 

13.  What  will  8  knives  cost,  at  6  dimes  each? 


22  MULTIPLICATION. 

14.  At  12  cents  each,  what  will  9  primers  cost? 

15.  At  10  dollars  each,  what  will  11  muffs  cost  ? 

16.  What  cost  8  oranges,  at  3  cents  apiece  ? 

17.  What  cost  7  melons,  at  9  cents  apiece? 

18.  What  cost  12  almanacs,  at  6  cents  each? 

19.  If  a  boat  moves  at  the  rate  of  11  miles  an 
hour,  how  far  will  it  go  in  11  hours  ? 

MODEL.  In  eleven  hours  it  will  go  11  times  as  many  miles  as  in 
1  hour.  If  in  1  hour  it  goes  11  miles,  in  11  hours  it  will  go  11 
times  11,  or  121,  miles.  Ans.  121  miles. 

20.  How  far  will  a  stage,  moving  at  the  rate  of  7 
miles  an  hour,  go  in  6  hours  ?    In  10  hours  ? 

21.  How  many  apples  in  5  baskets,  each  contain- 
ing 12  apples? 

22.  How  many  fingers  have  11  persons  ? 

23.  How  many  thumbs  have  8  persons  ? 

24.  If  a  boy  earns  $10  a  week,  how  many  dollars 
will  he  earn  in  4  weeks  ?    In  6  weeks  ?    In  9  weeks  ? 

25.  How  many  chickens  in  eleven  broods,  contain- 
ing nine  chickens  each  ? 

26.  Paid  $5  for  a  lamp,  and  $1  for  a  shade.     How 
much  will  four  such  lamps  and  shades  cost  ? 

27.  At  the  rate  of  7  dimes  for  a  knife,  and  2  dimes 
for  a  fork,  what  will  12  knives  and  forks  cost  ? 

28.  Four  pens,  each  containing  5  pigs,  hold  how 
many  pigs  in  all  ? 

29.  5x4=howmany? 

30.  5  +  5-ho  +  5=how  many? 

31.  Multiplication  is  a  short  way  of 'performing 
what  other  process  ?    Prove  this  in  the  case  of  six- 
times  three. 


MULTIPLICATION.  23 

SECTION  8.— 1.  How  much  is  twice  34  ? 
MODEL.     Twice  4  units  is  8  units ;  twice  3  tens  is  6  tens.     6  tens 
and  8  units  are  68.     Ans.  68. 

2.  How  much  is  3  times  23  ?     4  times  120  ? 

3.  How  much  is  twice  21  ?     311x3?     101  x  4  ? 

4.  How  much  is  twice  44  ?     22  x  4  ?     102  x  3  ? 

5.  How  much  is  4  times  1000  ?     9  times  100  ? 

6.  How  much  is  3  times  2000  ?     7  times  101  ? 

7.  How  much  is   10  times  7?     10  times  9?     10 
times  10?     10  times  100?     10  times  17? 

8.  How  do  we  multiply  a  number  by  10  ? 

9.  How  much  is  100  times  2  ?     100  times  9?     100 
times  20  ?     100  times  35  ?     100  times  68  ? 

10.  What  easy  way  is  there  of  multiplying  by  100  ? 

11.  How  much  is  20  times  7  ?     9  x  20  ?     14  x  20  ? 

12.  How  much  is  200  times  7  ?     200  times  9  ? 

13.  How  much  is  2000  times  7  ?     2000  times  9  ? 

14.  How  much  is  4  times  91  ? 

MODEL.  4  times  1  unit  is  4  units ;  4  times  9  tens  is  36  tens,  or  3 
hundreds  and  6  tens.  3  hundreds  6  tens  and- 4  units  are  364.  Ans. 
364. 

15.  How  much  is  5  times  61  ?     4  times  72  ? 

16.  How  much  is  twice  640  ?     8  times  61  ? 

17.  If  there  are   144  steel  pens  in  one  box,  how 
many  are  there  in  two  boxes  ? 

18.  How  many  pages  are  there  in  10  volumes,  con- 
taining 288  pages  each  ? 

19.  How  many  trees  has  a-  nurseryman,  if  he  has 
100  rows  of  17  each? 

20.  How  many  pills  are  there  in  200  boxes,  each 
containing  24  pills-? 


24  MULTIPLICATION. 

SECTION  9.— 1.  How  much  is  8  times  24  ? 

MODEL.  8  times  4  units  is  32  units,  or  3  tens  (which  we  add  to 
the  next  product)  and  2  units.  8  times  2  tens  is  16  tens,  and  3  tens 
make  19  tens.  19  tens  2  units  are  192.  Am.  192. 

2.  How  much  is  4  times  43  ?     32  x  6  ?     55  x  5  ? 

3.  How  much  is  9  times  24  ?     87x3?     98  x  2  ? 

4.  How  much  is  7  times  68  ?     14x8?     79  x  3  ? 

5.  How  much  is  5  times  103  ?     84  x  6  ?     38x9? 

6.  How  much  is  11  times  36?     12  times  51?     9 
times  803  ?     7  times  219  ?     12  times  45  ? 

7.  If  a  boy  learns  13  pages  every  week,  how  many 
pages  will  he  learn  in  12  weeks?    If  he  learns  12 
pages  a  week,  how  many  will  he  learn  in  13  weeks  ? 

8.  How  much  is  once  13  ?     Twice  13?     3  times 
13  ?    4  times  13  ?    5  times  13  ?    6  times  13  ?     7  times 
13  ?     8  times  13  ?     9  times  13  ? 

9.  How  much  is  13  times  1?     13  times  2?     13 
times  3?     13  times  4?     13  times  5?      13  times  6? 
13  times  7  ?     13  times  8  ?     13  times  9  ? 

10.  How  much  is  once  17?    Twice  17?     3  times 
17?    4  times  17?    5  times  17?    6  times  17?    7  times 
17  ?     8  times  17  ?     9  times  17  ? 

11.  How  much  is  17  times  1?     17  times  2?     17 
times  3?     17  times  4?     17  times  5?     17  times  6? 
17  times  7  ?     17  times  8  ?     17  times  9  ? 

12.  How  much  is  once  19  ?    Twice  19  ?     3  times 
19  ?     4  times  19  ?    5  times  19  ?    6  times  19  ?    7  times 
19?     8  times  19?     9  times  19  ? 

13.  How  much  is  19  times  1?     19  times  2?     19 
times  3  ?     19  times  4  ?     19  times  5  ?    19  times  6  ?    19 
times  7?     19  times  8  ?     19  times  9  ? 


MULTIPLICATION.  25 

14.  If  7  apples  can  be  bought   for  a  dime,  how 
many  can  be  bought  for  17  dimes  ? 

15.  A  person  bought  19  caps,  at  $3  apiece,  and 
paid  for  them  with  12  five-dollar  bills;  how  many 
dollars  should  he  receive  as  change  ? 

16.  How  much  will  a  boy  who  earns  $5  a  week, 
lack  of  having  earned  $100,  after  working  13  weeks? 

17.  How  many  fingers  have  thirteen  persons? 


SECTION  10.— Numbers  multiplied  together  are 
called  the  Factors  of  their  product. 

4  times  3  is  12 ;  4  and  3  are  factors  of  12. 

A  number  may  have  more  than  one  set  of  factors. 
4  times  3  is  12;  6  times  2  is  12;  12  has  two  sets  of 
factors.  What  are  they  ? 

1.  What  are  the  factors  of  8  ?     Of  15  ?     Of  22  ? 
Of  35?     Of  26?     Of  77?     Of  34? 

2.  Find  as  many  sets  of  factors  as  you  can  for  24. 
For  30.     For  36.     For  48.     For  16. 

3.  How  much  is  12  times  7  ? 

4.  (12=4x3)  How  much  is  4  times  7  ?  3  times  28? 

5.  (12  =  6  x  2)  How  much  is  6  times  7  ?  Twice  42  ? 

6.  Does  it  make  any  difference  whether  we  multi- 
ply by  a  product  at  once,  or  by  its  factors  in  turn  ? 

7.  How  much  is  18  times  8  ? 

8.  (18  =  9x2)  How  much  is  9  times  8  ?  Twice  72  ? 

9.  (18  =  3x6)  How  much  is  3  times  8  ?  6  times  24? 

10.  How  much  is  14  times  13  ? 

MODEL. — 14  being  7  times  2,  14  times  13  is  equal  to  7  times  13, 
multiplied  by  2.  7  times  13  is  91  ;  twice  91  is  182.  Ans.  182. 

a 


26  MULTIPLICATION. 

11.  How  much  is  18  times  23  ?     15  times  38  ? 

12.  How  much  is  16  times  26  ?     14  times  72  ? 

13.  How  much  is  21  times  17  ?     35  times  13  ? 

14.  How  much  is  25  times  19  ?     32  times  26? 
NOTE.     In  multiplying  numbers  together,  choose  the  easier  num- 
ber to  multiply  by. 

15.  If  one  sheep  yields  3   pounds   of  wool,  how 
many  pounds  will  45  sheep  yield  ? 

MODEL.  If  1  sheep  yields  3  pounds,  45  sheep  will  yield  45  times 
3  pounds.  45  times  3  equals  3  times  45,  or  135.  Ans.  135  pounds. 

16.  Twelve  make  a  dozen ;  how  many  are  23  dozen  ? 

17.  Twenty  make  a  score  ;  how  many  are  47  score  ? 

18.  How  many  are  three  score  years  and  ten  ? 

19.  Which  is  the  greater,  13  dozen  or  7  score  ? 

20.  How  many  dollars  will  an  acre  of  orange  trees 
produce  in  a  year,  if  they  average  $28  a  tree,  and 
there  are  53  trees  to  the  acre  ? 

MODEL.  If  one  tree  produces  $28,  53  trees  will  produce  53  times 
$28.  53  times  28=53  x  4  x  7.  Four  times  53  is  212  ;  seven  times 
212  is  1484.  Ans.  $1484. 

21.  What  cost  17  cows,  at  $56  each  ? 

22.  There   are   24  hours  in  one  day ;  how  many 
hours  in  23  days  ?     In  29  days  ? 

23.  How  many  trees  in  four  fields,  each  containing 
11  rows,  and  each  row  containing  9  trees  ? 

24.  How  many  letters  in  37  lines,  averaging  forty 
letters  to  the  line  ? 

25.  How  many  yards  in  14  pieces  of  cloth,  each 
containing  37  yards? 

26.  If  7  boys  have  each  6  hens,  and  each  hen  has 
8  chickens,  how  many  chickens  have  the  boys  in  all  ? 


MULTIPLICATION.  27 

27.  John  has  23  marbles ;  Samuel,  5  times  as  many. 
How  many  has  Samuel  ?    How  many  have  both  ? 

28.  Levi  has  16  cents  ;  Simon  has  7  times  as  many. 
How  many  have  both  ? 

29.  A  person  bought  6  baskets  of  fruit,  each  con- 
taining 8  peaches,  9  apples,  and  7  pears.     How  many 
peaches  had  he?    How  many  apples?    How  many 
pears  ?    How  many  of  all  three  ? 

30.  How  many  trees  in  13  fields,  each  containing 
6  apple  trees  and  9  pear  trees  ? 

31.  Every  day  a  boy  earned  90  cents,  and  spent 
65  ;  how  many  cents  had  he  at  the  end  of  six  days  ? 

32.  If  I  lay  in  200  pounds  of  butter,  and  use  13 
pounds  a  week  for  15  weeks,  how  much  will  remain? 

33.  Bought  23  cows  for  $48  each  ;  sold  them  at  $60 
apiece.  What  was  the  profit  on  each  ?  On  the  whole  ? 

34.  Sold,  at  $75  each,  18  lots  of  land  that  cost  $49 
apiece.     What  was  the  profit  on  the  whole  ? 

35.  A  ferryman  took  17  passengers  across  a  river 
for  5  cents  each,  and  then  lost  15  cents  of  what  they 
paid  him.     How  much  had  he  left  ? 

36.  How  many  flowers  in  10  bouquets,  each  con- 
taining 5  roses,  9  pinks,  and  7  daisies  ? 

37.  If  I  travel  19  miles  an  hour  for  6  hours,  and 
then  23  miles  an  hour  for  5  hours,  how  far  will  I  go 
in  the  whole  eleven  hours  ? 

38.  A  and  B,  travelling  toward  each  other,  met  in 
12  hours.     How  far  apart  were  their  starting-points, 
if  A  went  10  miles  an  hour,  and  B  12  ? 

39.  Find  first  the   sum,  then  the  difference,  and 
then  the  product,  of  10  and  35.     Of  11  and  36. 


28  DIVISION. 

40.  A  farmer,  having  40  acres  of  land,  gave  away 
15  of  them;  what  were  the  rest  worth,  at  $9  an  acre? 

41.  A  farmer,  having  85  acres  of  land,  gave  his 
son  and  daughter  each  25  acres.     What  was  the  land 
that  he  kept  worth,  at  $20  an  acre? 

42.  Sold  12  suits,  at  $17  for  each  coat,  $5  for  each 
vest,  and  $8  for  each  pair  of  pants.     What  was  re- 
ceived for  the  whole  ? 

43.  If  I  buy  6  almanacs  at  6  cents  apiece,  and  5 
more  at  5  cents  apiece,  and  sell  them  all  at  7  cents 
apiece,  what  is  my  profit  ? 

44.  How  much  is  12  times  7—7  times  12  ? 


CHAPTER     FOURTH. 

DIVISION. 

SECTION  11.— Division  is  the  process  of  finding 
how  many  times  one  number  is  contained  in  another. 

Division  is  denoted  by  a  short  horizontal  line  be- 
tween two  dots  -f-  ,  placed  after  the  number  to  be  di- 
vided. 6-f-2  is  read,  and  means,  six  divided  by  two. 

The  result,  or  number  obtained  by  dividing,  is 
called  the  Quotient.  6-^-2=3  ;  we  have  divided  6  by 
2,  and  3  is  the  quotient. 

One  number  is  not  always  contained  in  another  an 
exact  number  of  times.  Something  may  be  left  over, 
which  is  called  the  Remainder.  7-^2  =  3,  and  1  over; 
3  is  the  quotient,  and  1  the  remainder. 


DIVISION. 


29 


DIVISION  TABLE. 


1  in 

2  in 

3  in 

4  in 

1,    once. 

2,    once. 

3,    once. 

4,    once. 

2,    twice. 

4,    twice. 

6,    twice. 

8,    twice. 

3,    3  times. 

6,    3  times. 

9,    3  times. 

12,    3  times. 

j    4,    4  times. 

8,    4  times. 

12,    4  times. 

16,    4  times. 

'    5,    6  times. 

10,    5  times. 

15,    5  times. 

20,    5  times. 

6,    6  times. 

12,    6  times. 

18,    6  times. 

24,    6  times. 

7,    7  times. 

14,    7  times. 

21,    7  times. 

28,    7  times. 

8,    8  times. 

16,    8  times. 

24,    8  times. 

32,    8  times. 

9,    9  times. 

18,    9  times. 

27,    9  times. 

36,    9  times. 

10,  10  times. 

20,  10  times. 

30,  10  times. 

40,  10  times. 

11,  11  times. 

22,  11  times. 

33,  11  times. 

44,  11  times. 

12,  12  times. 

24,  12  times. 

36,  12  times. 

48,  12  times. 

5  in 

6  in 

7  in 

8  in 

5,    once. 

6,    once. 

7,    once. 

8,    once. 

10,    twice. 

12,    twice. 

14,    twice. 

16,    twice. 

15,    3  times. 

18,    3  times. 

21,    3  times. 

24,    3  times. 

20,    4  times. 

24,    4  times. 

28,    4  times. 

32,    4  times. 

25,    5  times. 

30,    5  times. 

35,    5  times. 

40,    5  times. 

30,    6  times. 

36,    6  times. 

42,    6  times. 

48,    6  times. 

35,    7  times. 

42,    7  times. 

49,    7  times. 

56,    7  times. 

40,    8  times. 

48,    8  times. 

56,    8  times. 

64,    8  times. 

45,    9  times. 

54,    9  times. 

63,    9  times. 

72,    9  times. 

50,  10  times. 

60,  10  times. 

70,  10  times. 

80,  10  times. 

55,  11  times. 

•66,  11  times. 

77,  11  times. 

88,  11  times. 

60,  12  times. 

72,  12  times. 

84,  12  times. 

96,  12  times. 

9  in 

10  in 

11  in 

12  in 

9,    once. 

10,    once. 

11,    once. 

12,    once. 

18,    twice. 

20,    twice. 

22,    twice. 

24,    twice. 

27,    3  times. 

30,    3  times. 

33,    3  times. 

36,    3  .times. 

36,    4  times. 

40,    4  times. 

44,    4  times. 

48,    4  times. 

45,    5  times. 

50,    5  times. 

55,    5  times. 

60,    5  times. 

54,    6  times. 

60,    6  times. 

66,    6  times. 

72,    6  times. 

63,    7  times. 

70,    7  times. 

77,    7  times. 

84,    7  times. 

72,    8  times. 

80,    8  times. 

88,    8  times. 

96,    8  times. 

81,    9  times. 

90,    9  times. 

99,    9  times. 

108,    9  times. 

90,  10  times. 

100,  10  times. 

110,  10  times. 

120,  10  times. 

99,  11  times. 

110,  11  times. 

121,  11  times. 

132,  11  times. 

108,  12  times. 

120,  12  times. 

132,  12  times. 

144,  12  times. 

SO  DIVISION. 

1.  How  many  times  is  4  contained  in  28  ?     5  in 
15?     6  in  24?     3  in  27  ?     2  in  16  ?     1  in  9?     5  in 
35?     6  in  36?     2  in  12  ?     3  in  36  ?     5  in  55  ? 

2.  How  many  times  is  7  contained  in  14  ?     9  in 
36?     10  in  40?     8  in  56  ?     11  in  33?     9  in  54  ?     12 
in  72?     7  in  70?     10  in  70  ?     11  in  99?     8  in  40  ? 

3.  How  many  times  is  2  contained  in  24?     12  in 
24?     3  in  18?     6  in  18  ?     12  in  108?     8  in  96  ?     9  in 
99?     11  in  121?     12  in  144?     10  in  120?     9  in  45  ? 

4.  How  many  times  is  3  contained  in  26  ?     (Ans.  8 
times,  and  2  over.)     6  in  43  ?     9  in  20  ?     2  in  11  ?    10 
in  95  ?     9  in  91  ?     5  in  43  ?     8  in  66  ?     4  in  14  ? 

5.  What  is  the  quotient,  and  what  the  remainder, 
in  the  following  ?     14-4-5?     68-r-7?     9-4-2?     39-^-4? 
112-4-11?    62-4-10?     61H-7?     51-4-8?     19—2?     63-4- 
5?     140-M2?     150-4-12?     133-4-11  ?     27-hlO? 

6.  If  6  tops  cost  30  cents,  what  will  1  cost  ? 
MODEL.     If  6  tops  cost  30  cents,  1  top  will  cost  as  many  cents 

as  6  is  contained  times  in  30,  or  5.     Ans.  5  cents. 

7.  If  4  lemons  cost  8  cents,  what  will  one  cost  ? 

8.  If  8  vests  cost  $32,  how  much  is  that  apiece  ? 

9.  What  will  1  primer  cost,  if  9  cost  81  cents  ? 

10.  If  7  hats  cost  $42,  what  will  one  hat  cost  ? 

11.  If  1  hat  cost  $6,  how  many  can  I  buy  for  $42  ? 

12.  $5  make  an  eagle  ;  how  many  eagles  in  $25  ? 

13.  At  10  cents  a  ride,  how  many  rides  can  a  per- 
son take  for  80  cents  ?    For  100  cents  ? 

14.  If  72  marbles  are  divided  equally  among  9 
boys,  how  many  marbles  will  each  receive  ? 

15.  If  it  takes  48  yards  of  calico  to  make  4  dresses, 
how  many  yards  will  it  take  for  1  dress  ? 


DIVISION.  31 

16.  How  many  rows  of  12  each  will  96  pins  make  ? 

17.  120  units  make  how  many  dozen  ? 

18.  At  what  rate  per  hour  is  a  steamboat  moving, 
en  it  goes  77  miles  in  7  hours  ? 

i£  How  many  ten-gallon  cans  will  be  required  to 
hold  50  gallons  of  milk  ? 

20.  How  many  benches,  holding  11  children  each, 
will  it  take  to  hold  88  children  ? 


SECTION  12,— 1.  What  is  the  product  of  4  and 
3?     12-7-4=how  many?     12-4-3— how  many ? 

2.  What  is  the  product  of  3  and  7  ?    How  many 
times  is  3  contained  in  21?     7  in  21? 

3.  What  is  the  product  of  5  and  4  ?    How  many 
times  is  5  contained  in  20  ?     4  in  20  ? 

4.  When  we  divide  a  product  of  two  factors  by 
one  of  the  factors,  what  do  we  get  ? 

5.  What  are  the  factors   of  22?      22-Ml=how 
many?     22 ~ 2 = how  many  ? 

6.  Divide  7  times  4  by  4.     Divide  7  times  4  by  7. 

7.  How  many  times  is  4  x  3  contained  in  12  x  11  ? 

8.  How  many  times  is  15  —  11  contained  in  13  x  4  ? 

9.  How  many  times  is  4  +  17  contained  in  9  x  21  ? 

10.  How  many  pigs,  at  $9  apiece,  should  a  man 
give  in  exchange  for  9  sheep  at  $8  apiece  ? 

11.  A  father  buys  ten  dimes'  worth  of  oranges,  at 
the  rate  of  three  for  a  dime,  and  divides  them  equally 
between   his   daughter    and  two   sons ;    how  many 
oranges  does  each  receive  ? 


32  DIVISION. 

12.  A  makes  $6  a  day  for  5  days.     B  buys  some 
goods  for  $48,  and  sells  them  for  $53.    How  many  such 
lots  of  goods  must  B  buy  and  sell,  to  make  as  much  as  A  ? 

13.  How  many  times  is  10  contained  in  30  ?     In 
50?     In  110?     In  60?     In  10  ?     In  70  ? 

14.  How  do  we  divide  a  number  by  10  ?    Am.  By 
cutting  off  its  last  figure. 

15.  10  in  100,  how  many  times  ?     10  in  140  ?     10 
in  230?     10  in  360?     10  in  780  ?     10  in  970  ? 

16.  If  the  figure  thus  cut  off  is  not  0,  what  do  we 
call  it?    Ans.  Remainder;    10   in  75  is  contained  7 
times,  and  5  remainder. 

17.  10  in  105,  how  many  times  ?     10  in  174  ?     10 
in  211?     10  in  863?     10  in  329?     10  in  468  ? 

18.  How  do  we  divide  a  number  by  100?    Ans. 
By  cutting  off  its  last  two  figures. 

19.  How  many  times  is  100  contained  in  200  ?    In 
700  ?     In  1000  ?     In  1300  ?     In  1500  ?     In  3000  ? 

20.  If  the  figures  thus  cut  off  are  not  naughts, 
what  do  we  call  them?      Ans.  Remainder;    100  in 
354  is  contained  3  times,  and  54  remainder. 

21.  How  many  times  is  100  contained  in  109  ?    In 
483  ?     In  1007  ?     In  1708  ?     In  1780  ?     In  9657  ? 

22.  If  30  loaves  are  divided  equally  among  10  poor 
families,  how  many  loaves  will  each  receive  ? 

23.  If  $1100  is  to  be  raised  in  equal  parts  from  100 
persons,  how  many  dollars  must  each  pay  ? 

24.  If  3  boys  and  7  girls  obtain  1200  good  marks 
during  a  term,  what  is  the  average  to  each  ? 

25.  How  many   hundred-dollar  watches   can   be 
bought  for  $1900  ?     For  $2500  ?     For  $2999  ? 


DIVISION.  33 

SECTION  13.-— 1.  3  in  6390,  how  many  times  ? 

MODEL.  3  is  contained  in  6  thousands,  2  thousand  times ;  in  3 
hundreds,  1  hundred  times  ;  in  9  tens,  3  tens,  or  thirty  times  ;  hi  0 
units,  0  times.  Ans.  2130  times. 

2.  How  many  times  13  4  contained  in  844  ?     5  in 
550  ?     2  in  68  ?     3  in  936  ?     4  in  480  ?     2  in  246  ? 

3.  How  many  times  is  9  contained  in  819  ? 
MODEL.     9  is  not  contained  in  8.     In  81  tens  it  is  contained  9 

tens,  or  90  times;  in  9,  once.     Am.  91. 

4.  How  many  times  is  8  contained  in  648  ?     7  in 
4277?     6  in  5406?     4  in  3284?     3  in  2793  ? 

5.  Find  the  quotient.     1082-7-2.     1869-e-3.     4860 
-4-6.     4055-4-5.     128-7-4.     3690-4-9.     2408-4-8. 

6.  How  many  times  is  7  contained  in  224  ? 
MODEL.     7  is  not  contained  in  2.     7  in  22  tens,  3  tens,  or  30 

times,  and  1  ten  over.     1   ten  and  4  units  are  14  units ;  7  in  14, 
twice.     Ans.  32. 

7.  How  many  9's  in  207  ?     In  153  ?     In  288  ?     In 
414  ?     In  171  ?     In  468  ?     In  243  ?     In  747  ? 

8.  576  is  how  many  times  6  ?    How  many  times 
8  ?    How  many  times  12  ?     How  many  times  4? 

9.  In  how  many  hours  can  a  boat  go  192  miles,  if 
it  moves  at  the  rate  of  8  miles  an  hour? 

10.  After  sailing  56  miles,  how  long  will  it  take 
this  boat  to  perform  the  rest  of  its  trip  of  192  miles? 

11.  How  many  dozen  oranges  can  be  sold  out  of  a 
load  containing  372  oranges? 

12.  How  many  bags,  containing  8  pecks  each,  will 
be  needed  to  hold  456  pecks  of  oats  ? 

13.  There  being  7  days  in  1  week,  how  many  weeks 
are  there  in  364  days  ? 


34  DIVISION. 

14.  If  a  person's  expenses  amount  to  $11  a  week, 
how  many  weeks  will  $319  last  him  ? 

15.  A  girl  is  fastening  buttons  on  cards  in  rows  of 
12  each.     How  many  rows  will  she  make  out  of  400 
buttons,  and  how  many  buttons  over  ? 

16.  By  selling  6  barrels  of  flour  for  $69,  a  man 
made  $3.     What  did  his  flour  cost  him  a  barrel  ? 

17.  If  there  are  3  sheets  used  in  printing  a  pam- 
phlet, how  many  pamphlets  will  1086  sheets  make? 

1 8.  Allowing  5  beets  to  a  bunch,  how  many  bunches 
will  493  beets  make,  and  how  many  beets  over  ? 

19.  A  person  drew  $765  out  of  the  bank  in  five-dol- 
lar bills ;  how  many  bills  had  he  ?     How  many  of 
these  bills  must  he  use,  to  pay  a  debt  of  $95  ? 


SECTION  14. — 1.  2  in  20,  how  many  times  ?     20 
in  20  ?     2  in  40,  how  many  times  ?     20  in  40  ? 

2.  3  in  60,  how  many  times  ?     30  in  60  ?     3  in  90, 
how  many  times  ?     30  in  90  ?     30  in  120  ? 

3.  40  in  80,  how  many  times  ?     In  160  ?     In  200  ? 
50  in  200,  how  many  times  ?     In  250  ?     In  300  ? 

4.  How  do  we  divide  by  40?     Ans.  By  cutting 
off  the  last  figure  of  the  number  to  be  divided,  and 
then  dividing  by  4,  prefixing  the  remainder,  if  any,  to 
the  figure  cut  off,  for  the  true  remainder. 

5.  How  do  we  divide  by  50  ?      How  do  we  di- 
vide by  80  ?     By  90  ? 

6.  How  do  we  divide  by  60  ?    By  70  ? 

7.  How  many  times  is  60  contained  in  360  ?     In 
480  ?     In  660  ?     In  720  ?     In  2400  ?     In  1200  ? 


DIVISION.  35 

8.  70  is  contained  in  140  how  many  times  ?     In 
490?     In  840?     In  2100?     In  770  ?     In  280  ? 

9.  80  in  80  ?     80  in  160  ?     80  in  400  ?     80  in  720  ? 
80  in  2400  ?     80  in  480  ?     80  in  640  ? 

10.  90  in  90  ?     90  in  270  ?     90  in  450  ?     90  in  630  ? 
90  in  810  ?     90  in  1800  ?     90  in  3600  ? 

11.  How  many  pounds  of  meat,  at  20  cents  a  pound, 
can  be  bought  for  180  cents? 

12.  How  many  rows  of  30  each  will  930  trees  make  ? 

13.  How  many  50-dollar  bills  make  $1000  ? 

14.  How  many  100-dollar  bills  make  $3000  ? 

15.  How  many  60-acre farms  will  2400  acres  make? 

16.  How  many  70-dollar  watches  will  $840  buy? 

17.  How  many  mules,  at  $100  each,  can  I  buy  for 
$1235,  and  how  much  will  be  left  over? 

18.  How  many  years  will  it  take  a  certain  house  to 
yield  me  $960,  if  I  let  it  for  $120  a  year  ? 

19.  Into  how  many  gangs  of  40  men  can  280  la- 
borers be  divided  ? 

20.  A  drover,  having  720  sheep,  divides  them  into 
flocks  of  80  each ;  how  many  flocks  does  he  make  ? 
After  selling  5 'of  these  flocks,  how  many  sheep  has  he 
left  ?     What  are  they  worth,  at  $8  each  ? 

21.  In  how  many  days  will  a  man  travel  2000  miles, 
if  he  travels  100  miles  a  day  ?     If  200  a  day  ? 

22.  If  a  person  lays  up  $200  a  year,  in  how  many 
years  will  he  be  worth  $4000  ?     $8000  ? 

23.  In  a  certain  library  are  2400  volumes.     How 
many  cases  will  be  needed  to  hold  them,  allowing  10 
shelves  to  each  case  and  30  volumes  to  each  shelf? 

24.  Divide  1200  x  2  by  700—400. 


36  DIVISION. 

SECTION  15.— 1.  What  is  the  result  of  addition 
called  ?  Of  subtraction  ?  Of  multiplication  ?  Of 
division  ? 

2.  What  operation  is  performed,  to  produce  a  quo- 
tient ?    A  sum  ?    A  product  ?     A  difference  ? 

3.  Find  first  the  product,  then  the  quotient,  then 
the  sum,  and  then  the  difference,  of  5  and  105  ? 

4.  How  many  times  is  the  difference  between  75 
and  105  contained  in  their  sum  ? 

5.  How  many  times  is  132 -r- 11  contained  in  72x4? 

6.  A  boy  caught  25  chub,  56  perch,  13  pout,  and 
1 1  eels.     If  he  divided  them  equally  with  four  of  his 
friends,  how  many  fish  had  each  ? 

7.  From  New  York  to  Albany,  the  distance  is  144 
miles;  from  New  York  to  Sing  Sing,  by  the  same 
road,  30  miles.    How  long  will  it  take  to  go  from  Sing 
Sing  to  Albany,  at  the  rate  of  12  miles  an  hour? 

8.  A  borrows  $238  from  B;  how  much  a  month 
must  he  give  B,  to  pay  the  debt  in  12  months? 

9.  A  farmer  had  18  ducks,  12  turkeys,  11  geese,  39 
chickens,  and  10  guinea-fowls.     After  selling  34  of  the 
whole  number,  he  put  the  rest  in  4  coops,  dividing 
them  equally ;  how  many  were  in  each  coop  ? 

10.  How  many  pounds   of  butter,  at  40  cents  a 
pound,  should  be  given  for  20  pounds  of  meat,  worth 
18  cents  a  pound  ? 

11.  A  person  invested  $960  in  apples,  at  $3  a  bar- 
rel.    How  many  trees,  at  the  rate  of  2  barrels  to  a 
tree,  did  it  take  to  produce  these  apples  ? 

12.  How  long  will  8000  pounds  of  flour  last  25 
men,  allowing  each  man  4  pounds  a  day  ? 


DIVISION.  3  7 

13.  Six  boys  put  in  40  cents  each,  and  bought  some 
melons,  at  12  cents  apiece ;  how  many  did  they  buy? 

14.  A  person,  having  to  make  a  journey  of  154 
miles,  travelled  13  miles  an  hour  for  4  hours,  and  14 
miles  an  hour  for  3  hours.     How  many  hours  will  it 
take  him  to  complete  his  journey,  travelling  the  rest 
of  the  way  12  miles  an  hour? 

15.  How  many  boxes,  containing  20  pens  each,  can 
a  person  fill  from  5  large  boxes,  holding  144  each  ? 

16.  A  boy  wishes  to  buy  a  pony  for  $100.     If  he 
lays  up  $8  a  month  for  12  months,  how  much  will  he 
lack? 

17.  If  you  owe  a  person  $50  for  one  bill  of  goods 
and  $47  for  another,  how  much  of  the  whole  debt  can 
you  pay  with  10-dollar  bills  ? 

18.  If  11  men  can  do  a  job  in  33  days,  how  long 
will  it  take  1  man  to  do  it  ?     How  long,  3  men  ? 

19.  Two  men  travel  from  the  same  place  in  oppo- 
site  directions,  one   10    miles  an  hour,  the  other  6 
miles  an  hour ;  how  far  apart  are  they  at  the  end  of 
11  hours? 

20.  A  house  was  bought  for  $1200,  and  sold  for 
$1500.     The  profit  was  divided  between  4  persons  ; 
what  was  the  share  of  each  ? 

21.  Laura  has  17  chestnuts,  twice  as  many  hickory 
nuts,  and  as  many  peanuts  as  she  has  chestnuts  and 
hickory  nuts.     If  she  divides  them  all  among  6  of  her 
playmates,  how  many  nuts  does  each  receive  ? 

22.  Twenty  acres  of  land  are  bought  for  $80.  How 
many  dollars  per  acre  must  they  be  sold  for,  that  the 
purchaser  may  double  his  money  ? 


FRACTIONS. 


Ans.  By 
Ans.  By 


CHAPTER    FIFTH. 

FRACTIONS. 

SECTION  16.— 1.  If  we  divide  a  pear 
into  two  equal  parts,  what  is  each  part 
•called?  Ans.  A  half. 

2.  How  many  halves  in  a  whole  ? 

3.  How  do  we  get  half  of   anything? 

•  dividing  it  into  2  equal  parts. 

4.  How  do  we  find  half  of  a  number  ? 

•  dividing  it  by  2. 

5.  What  is  half  of  6  ?     Of  10  ?     Of  18  ?     Of  60  ? 
*Of2?     Ofl?     Of  80?     Of  100?     Of  1000? 

6.  If  we  divide  a  pear  into  three 
equal  parts,  what  is  each  part  called? 
Ans.  One  third. 

7.  How  many  thirds  in  a  whole  ? 

8.  How  do  we  get  a  third  of  anything  ? 

9.  How  do  we  find  one  third  of  a  number  ?     Ans. 
By  dividing  it  by  3. 

"  10.  What  is  "one  third  of  12?      Of  15?     Of  27? 
'•Of  36?     Of  42?     Of  3?     Of  300?     Ofl? 

11.  If   we   divide   a   pear  into  four 
•equal  parts,  what  is  each  part  called? 
.Ans.  One  fourth,  or  quarter. 

12.  How  many  fourths  in  a  whole? 

13.  How  many  quarters  in  1  ? 

,     14.  How  do  we  get  a  fourth  of  anything  ? 

15.  -How  do  we  find  one  fourth,  or  quarter,  of  a 
.number?     Ans.  By  dividing  it  by — what? 


FRACTIONS.  39 

16.  How  much  is  one  fourth  of  4  ?     Of  1  ?     Of 
44?     Of  88?     Of  96?     One  quarter  of  100?    Of  200? 

17.  If  we  divide  a  pear  into  five  equal 
parts,  what   is  each  part   called?     Ans. 
One  fifth. 

18.  How  many  fifths  in  a  whole,  or  1  ? 

19.  How  do  we  find  one  fifth  of  a  number? 

20.  How  much  is  one  fifth  of  25  ?     Of  95  ?     Of 
125  ?     Of  1  ?     Of  200  ?     Of  300  ?     Of  3000  ? 

21.  If  we  divide  a  pear  into  six  equal 
''parts,  what  is   each  part  called?      Ans. 

One  sixth. 

22.  How  many  sixths  in  1  ? 

23.  How  do  we  find  one  sixth  of  a  number  ? 

24.  How  much  is  one  sixth  of  6  ?     Of  1  ?     Of  72  ? 
Of  120?     Of  366?     Of  480?     Of  612? 

25.  If  we  divide  a  whole  into  seven  equal  parts, 
what  is  each  part  called  ?    Ans.  One  seventh. 

26.  How  many  sevenths  in  1  ? 

27.  Howdo  we  find  one  seventh  of  a  number? 

28.  How  much  is  one  seventh  of  63  ?     Of  91  ?    Of 
112?     Of  140?     Of  280?     Of  700?     Of  350  ? 

29.  If  we  divide  a  whole  into  eight  equal  parts, 
what  is  each  part  called  ?    Ans.  One  eighth. 

30.  If  we  divide  a  whole  into  nine  equal  parts, 
what  is  each  part  called  ?    Ans.  One  ninth. 

31.  How  do  we  get  tenths,  elevenths,  &c.  ?    Ans. 
By  dividing  a  whole  into  10,  11,  &c.,  equal  parts. 

32.  How  do  we  get  thirteenths  ?  Fifteenths?  Thir- 
tieths ?     Thirty-seconds  ?     Forty-firsts  ? 


40  FRACTIONS. 

33.  How  many  twelfths  in  1  ?     How  many  nine- 
teenths?   How  many  twenty-firsts? 

34.  How  do  we  find  one  eighth  of  a  number  ?  How 
do  we  find  one  tenth  ?  One  thirteenth  ?  One  thirtieth  ? 

35.  How  much  is  one  eighth  of  72  ?     Of  32  ?     Of 
96  ?     One  ninth  of  117  ?     Of  135  ?     One  tenth  of  20  ? 
Of  200  ?     Of  2000  ?     One  eleventh  of  132  ?     Of  220  ? 
Of  1650  ?     One  twelfth  of  168  ? 

36.  How  much  greater  is  one  twelfth  of  600  than 
one  twentieth  of  100  ? 


SECTION  17—1.  What  are  one  halftone  third, 
one  fourth,  &c,,  called  ?     Ans.     Fractions, 

2.  Into  what  is  the  pear  on  the  left  divided  ?    Into 

what,   the    pear  on  the   right  ?< 
Which  is  greater,   one  half  or 
one  third  ? 

3.  Which  is  greater,  one  half  a  line  or  one  fourth 
of  the  same  line  ? 


4.  The  more  parts  we   divide  a  thing  into,   the 
larger  or  smaller  must  those  parts  be  ? 

5.  Walter   divides   a   quart  of  chestnuts  equally 
among  his  3  brothers ;  what  part  of  a  quart  will  each 
have  ?    He  then  divides  a  quart  equally  among  his 
4  sisters ;    what  part  will  each  have  ?     Which  Avill 
have  the  larger  share,  one  of  the  brothers  or  one  of 
the  sisters  ? 

6.  Which  is  greater,  one  fifth  or  one  sixth  ? 


FRACTIONS.  41 

7.  How  much  is  one  fifth  of  60  ?     One  sixth  of 
60  ?     Which  is  the  greater  ? 

8.  If  A  can  dig  a  certain  ditch  in  12  days,  what 
part  of  it  can  he  do  in  1  day? 

9.  If  B  can  dig  the  same  ditch  in  10  days,  what 
part  can  he  dig  in  1  day  ?     Can  he  do  more  or  less 
than  A  ? 

10.  If  a  pipe  can  empty  a  cistern  in  7  hours,  what 
part  of  the  cistern  can  it  empty  in  one  hour? 

11.  If  a  person  leaves  his  property,  worth  $1800, 
to  his  wife  and  8  children,  in  equal  parts,  what  fraction 
of  the  whole  will  each  have,  and  how  many  dollars  ? 

12.  Four  equal  partners  make  a  profit  of  $2400; 
what  part  of  the  whole,  and  how  much,  must  each  re- 
ceive ? 

13.  If  you  put  up  ten  pounds  of  tea  in  packages 
of  one  pound,  what  part  of  the  whole  do  you  put  in 
each  package  ? 

14.  1  is  what  part  of  10  ?     Of  5  ?     Of  15  ?     Of  9  ? 

15.  The  earth  turns  on  its  axis  in  24  hours ;  what 
part  of  a  revolution  does  it  perform  in  1  hour? 

16.  There  being  100  cents  in  a  dollar,  what  part  of 
a  dollar  is  1  cent  ? 

17.  If  a  barrel  of  flour  will  last  a  family  30  days,, 
what  part  of  it  will  last  them  one  day  ? 

18.  One  is  what, part  of  9  times  11  ? 

19.  How  many  fifths  make  1  ?     How  many  ninths  ? ' 

20.  Which  is  greater,  5  fifths  or  9  ninths  ? 

21.  A  person  having  $85,  earned  $115  more,  and 
then  gave  away  one  fourth  of  what  he  had.     How 
much  did  he  give  away  ?     How  much  did  he  retain  ? 


42  FRACTIONS. 

SECTION  18.— 1.  If  we  divide  an  orange  into  4 
equal  parts,  what  is  one  such  part  called  ?     What  are 

2  such  parts  called  ?     Ans.  Two  fourths.     What  are 

3  such  parts  called  ? 

2.  If  you  divide  25  cents  equally  among  5  beggars, 
what  part  will  one  beggar  receive?     Two  beggars? 
Three  beggars  ?     Four  beggars  ?     What  part  will  ail 
five  beggars  receive  ?     Ans.  5  fifths,  or  the  whole. 

3.  What  part  of  10  is  1  ?     What  part  of  10  is  2  ? 
MODEL.     1  is  1  tenth  of  10 ;  and  2  is  twice  1  tenth,  or  2  tenths. 

Ans.  2  is  2  tenths  of  10. 

4.  What  part  of  11  is  1  ?     What  part  of  11  is  3  ? 

5.  What  part  of  12  is  1  ?     What  part  of  12  is  5? 

6.  What  part  of  9  is  7  ?     8  is  what  part  of  13  ?   15 
is  what  part  of  19  ?     What  fraction  of  100  is  21  ? 

7.  If  we  divide  one  by  3  (that  is,  divide  1  into  3 
equal  parts),  what  is  the  quotient  ?     Ans.  One  third. 

8.  If  we  divide  2  by  3,  what  is  the  quotient  ?   Ans. 
f  wo  thirds. 

9.  If  we  divide  1  by  5,  what  is  the  quotient  ?     2 
by  5  ?     3  by  5  ?     4  by  5  ? 

10.  If  we  divide  7  by  5,  what  is  the  quotient,  and 
what  the  remainder  ? 

11.  What  may  we  call  this  remainder,  2  ?     Ans. 
As  the  2  is  to  be  divided  by  5,  we  may  call  it  2  fifths. 

12.  If  we  divide  11  by  8,  what  is  the  quotient? 
Ans.  1  and  3  eighths. 

13.  How  many  times  is  4  contained  in  23  ?    Ans. 
5  and  3  fourths. 

14.  (Express  the  remainder  as  above.)     53-^-7  = 
how  many?     99-h8?     46^4?     37-hlO?     120—11? 


FRACTIONS.  43 

SECTION  19,— 1.  What  is  a  Fraction  ? 
Ans.  A  Fraction  is  one  or  more  of  the  equal  parts 
into  which  a  whole  is  divided. 

2.  Learn  how  fractions  are  written. 


One  half  £ 

One  third  i 

One  fourth  J 

One  fifth  £ 

One  sixth  ^ 

One  seventh  ^ 

One  eighth  J 

One  nijith  ' 


One  tenth 
One  eleventh 
One  twelfth 
Two  twelfths 
Three  twelfths 
Four  twenty-firsts 
Five  thirty-seconds 
Six  hundredths 


T2 


T2 


3.  A  written  fraction  consists  of  two  numbers,  one 
below  the  other,  with  a  line  between. 

The  number  below  the  line  is  called  the  Denomi- 
nator. It  shows  into  how  many  equal  parts  the 
whole  is  divided,  and  gives  name  to  these  parts. 

The  number  above  the  line  is  called  the  Numera- 
tor. It  shows  how  many  of  the  equal  parts  denoted 
by  the  Denominator  are  taken. 

The  Numerator  and  the  Denominator,  taken  to- 
gether, are  called  the  Terms  of  the  fraction. 

g  is  a  fraction.  5  and  9  are  its  Terms.  9  is  the  Denominator, 
and  shows  that  the  whole  is  divided  into  nine  equal  parts,  making 
each  part  one  ninth.  5  is  the  Numerator,  and  shows  that  five  of 
these  equal  parts  are  taken.  In  reading,  name  the  numerator  first, 
and  pronounce  ths  distinctly  at  the  end,— ^&>e  ra'wTHS. 

4.  Read  each  fraction ;  name  its  terms ;  mention 
its  numerator  and  denominator  ;  tell  what  each  shows. 

i*     f-     fi*     TP     !&•     W-     Tlr- 


44  FRACTIONS. 

5.  Write  in  figures  : — Two  fifths.     Three  sevenths. 
Five  sixths.     Seven  elevenths.     Five  nineteenths. 

6.  Write  in  figures : — Seven  tenths.     Seven  hun- 
dred ths.     Seven  thousandths.     Fifty-one  thousandths. 
Xineteen  sixtieths.     Four  seventy-fifths. 

7.  How  much  is  ^  of  18  ?     How  much  is  |  of  18  ? 

o  o 

MODEL.     One  third  of  18  is  6,  and  two  thirds  are  twice  6,  or  12. 
Ans.  12. 

8.  How  much  is  f  of  12  ?     £  of  10  ?.   f  of  20  ? 

9.  How  much  is  f  of  18  ?     f  of  21  ?     -f  of  35  ? 

10.  How  much  is  T9¥  of  100  ?   •  f  of  36  ?    -f  of  77  ? 

11.  How  much  is  -fr  of  121  ?     £  of  40  ?     |  of  54  ? 

12.  How  much  is  -&  of  144  ?     f  of  60  ?     f  of  72  ? 

13.  How  much  is  ^  of  160  ?     f  of  45  ?     -f  of  88  ? 

14.  How  much  is  J±-  of  90  ?     f  of  50  ?     f  of  91  ? 

15.  How  much  is  Jf  of  40  ?     T^  of  96  ?     T6T  of  99  ? 

16.  How  much  greater  is  T7¥  of  80  than  f  of  48  ? 

17.  How  much  less  is  f  of  35  than  T5T  of  33  ? 

18.  Which  is  greater,  -^  of  108  orf  of  63  ? 

19.  A  person  having  $1550   divided   it   into  five 
equal  parts,  and  gave  one  of  these  parts  to  each  of  his 
three  children.     What  part  of  the  whole  did  the  chil- 
dren receive,  and  how  much? 

20.  A  road  160  miles  long  is  composed  of  10  equal 
sections.     What  fraction  of  the  whole  road  are  6  of 
these  sections,  and  how  many  miles  do  they  contain? 

21.  Two  brothers  are  equal  heirs  with  6  other  per- 
sons to  an  estate  of  $8000.     What  part  of  the  whole 
do  the  two  brothers  receive,  and  how  much  money? 

22.  If  the  rent  of  a  house  for  30  days  is  $90,  how 
much  will  it  be  for  1 7  days  ? 


FRACTIONS.  45 

SECTION  20.— 1.  What  is  a  Fraction  ? 

2.  What  does  one  third  mean  ?     Ans.  One  of  three 
equal  parts  into  which  a  whole  is  divided. 

3.  WThat  does  two  thirds  mean  ?    Ans.   Two  of 
three  equal  parts  into  which  a  whole  is  divided. 

4.  What  does  one  fourth  mean  ?   Two  fourths  ?  f  ? 

5.  How  is  this  expression,  6  J,  read  ?     Ans.  Six  and 
three  fourths. 

6.  Of  what  does  it  consist  ?      Ans.  Of  a  whole 
number  (6),  and  a  fraction  (f). 

7.  What  is  a  number  that  consists  of  a  whole  num- 
ber and  a  fraction  called  ?    Ans.  A  Mixed  Number. 

8.  How  many  fourths  in  6  ? 

MODEL.     In  1  there  are  4  fourths,  and  in  6  six  times  4  fourths, 
or  24  fourths.     Ans.  ^. 

9.  How  many  fourths  in  6|  ? 

MODEL.     In  1  there  are  4  fourths,  and  in  6  six  times  4  fourths, 
or  24  fourths.     24  fourths  and  3  fourths  are  27  fourths.     Ans.  *f. 

10.  How  many  eighths  in  7  ?     In  7f  ?     In  9|  ? 

11.  How  many  fifths  in  11  ?     In  11^?     In  12|? 

12.  How  many  ninths  in  12  ?     In  12£?     In  llf  ? 

13.  How  many  sevenths  in  4  ?     In  4-f  ?     In  8-|  ? 

14.  How  many  tenths  in  10  ?     In  10-^  ?     In  4^  ? 

15.  How  many  twelfths  in  6  ?     In  5TV  ?     In  8T\? 

16.  5  =how  many  halves?     5 J  ?     6£?     8£? 

17.  How  many  thirds  in  3  ?     In  7  ?     In  9  ?     In  5  ? 

18.  8£=how  many  sixths?     9|  ?     8|  ?     4f  ?     4  — 
how  many  elevenths  ?     4T2T  ?     7T3T  ?     9^  ?     6T\  ? 

19.  How   many   halves   in    50^  ?     Twentieths   in 
5£l?   Hundredths  in2Ti7?   Fortieths  in  4-4%  ?   Thirds 
in  13^  ?     Eighths  in  14|  ?     Quarters  in  7J  ? 


46  REDUCTION    OF   FRACTIONS. 

20.  What  kind  of  fractions  are  |5  £,  &c.  ?     Ans. 
Improper  Fractions. 

21.  What  is  an  Improper  Fraction  ?     Ans.  A  Frac- 
tion whose  numerator  is  equal  to,  or  greater  than,  its 
denominator. 

22.  What  is  a  fraction  called,  whose  numerator  is 
less  than  its  denominator  ?    Ans.  A  Proper  Fraction. 


SECTION  21.— 1.  How  many  thirds  do  4j  equal  ? 

2.  When  you  say  *£-  for  4J,  do  you  change  the  form 
of  the  mixed  number  ?     Do  you  change  its  value  ? 

3.  What  is  the  process  of  changing  the  form,  with- 
out changing  the  value,  called  ?    Ans.  Reduction. 

4.  In  the  last  Section,  what  kind  of  numbers  did 
you  reduce  ?     To  what  kind  of  fractions  did  you  re- 
duce  them  ? 

5.  5  units  equal  how  many  thirds  ?     15  thirds,  then, 
equal  how  many  units  ? 

6.  How  many  units  in  16  thirds  ? 

MODEL.     Since  3  thirds  make  1  unit,  16  thirds  make  as  many 
units  as  3  thirds  are  contained  times  in  16  thirds,  or  5J-.     Ans.  5^. 

7.  How  many  units  in  ^  ?     In  ^  ?    In  *£•  ? 

8.  How  many  units  in  ^-  ?     In  *£-  ?     In  -\5  ? 

9.  How  many  units  in  *$-  ?     In  ^  ?     In  ^  ? 

10.  How  many  wholes  in  i£  ?     In  ^  ?     In  f£  ? 

1 1.  How  many  wholes  in  f|  ?     In  fj  ?     In  f|  ? 

12.  Which  is  greater,  |J-  or  fj?    ^  or  -4/? 

13.  Which  is  greater,  ^  Or  \3-  ?     \°f  or  ¥  ? 

14.  How  many  times  1  in  -4/  ?     In 

15.  How  many  times  1  is 


REDUCTION    OF    FRACTIONS.  47 

16.  |£=how  many  units  ?     f  |  ?     -f f  ?     -^-  ? 

17.  Which  is  greater,  \5-  or  -5/  ?    -6T2-  or  -%*-  ? 

18.  How  many  dollars  in  ^  of  a  dollar?     In  29 
quarter-dollars  ?     In  29  half-dollars  ? 

19.  In  this  Section,  what  kind  of  fractions  have 
you  reduced  ? 

20.  To  what  have  you  reduced  them  ? 


SECTION  22, — 1.  What  is  a  Fraction  ?    A  Proper 
Fraction  ?     An  Improper  Fraction  ? 

2.  What  is  a  Mixed  Number  ? 

3.  What  is  meant  by  the  Reduction  of  Fractions  : 

4.  How  many  halves  in  ]  2  ?   How  many  units  in  %£-  ? 

5.  How  do  you  reduce  units  to  halves  ?     How  do 
you  reduce  halves  to  units  ? 

6.  How  do  you  reduce  thirds  to  units  ?     Units  to 
thirds  ? 

7.  How  do  you  reduce  units  to  sixths  ?    How  do 
you  reduce  sixths  to  units  ? 

8.  How  do  you  reduce  ninths  to  units  ?     Units  to 
ninths  ? 

9.  Reduce  8  to  twentieths.     Reduce  J-f-2-  to  units. 

10.  Reduce  ±£-  to  a  mixed  number.     Reduce  2f  to 
an  improper  fraction. 

11.  At  a  dinner,  47  plates  of  pie  were  called  for; 
how  many  pies   were  used,  if  each  plate  contained 
quarter  of  a  pie  ? 

12.  A  druggist  is  putting  up  soda  in  powders  of  \ 
of  a  dram  each.     If  he  uses  9|-  drams  of  soda,  how 
many  powders  does  he  put  up  ? 


48  FRACTIONS. 

13.  How  many  bows,  requiring  ^  of  a    yard  of 
ribbon  each,  can  be  made  out  of  7f  yards  ? 

14.  At  a  lecture  43  dollars,  24  half-dollars,  and  32 
quarter-dollars,  were  taken  in.     How  much  in  all  was 
received  ? 

1 5.  A  pint  is  one  eighth  of  a  gallon  ;  how  many  pints 
in  13f  gallons  ?     How  many  gallons  in  141  pints  ? 

16.  If  a  boy  can  saw  and  split  y1^  of  a  cord  of  wood 
in  a  day,  how  many  such  boys  will  it  take  to  saw  and 
split  4T\  cords  in  a  day  ? 

17.  How  many  cords  would  the  boy  just  mentioned 
saw  and  split  in  300  days  ? 

18.  How  many  dollars  will  53  tops  cost,  at  the  rate 
of  -£Q  of  a  dollar  apiece  ? 

19.  How  many  quarter-pound  weights  will  it  take 
to  balance  56  pounds  ?  How  many  half-pound  weights  ? 

20.  If  an  apple  costs  -^  of  a  dollar,  how  many 
apples  can  you  buy  for  $1  ?     For  $2  ?     For  $5  ? 

21.  How  many  lots  -^  of  an  acre  in  size  can  be  laid 
out  from  5  acres  ?     From  7  acres  ?     From  2T3T  acres  ? 

22.  How  many  inches  in  ^J-  of  an  inch  ? 

23.  One  day  is  %  of  a  week;  how  many  weeks  in 
50  days  ? 

24.  How  much  is  |  of  18  ?    1 4  is  how  many  times  2  ? 

25.  |  of  18  is  how  many  times  2  ? 

26.  |  of  60  is  how  many  times  9  ? 

27.  -|  of  56  is  how  many  times  8  ? 

28.  ^  of  50  is  how  many  times  15  +  5? 

29.  |  of  72  is  how  many  times  17—8  ? 

30.  |  of  84  is  how  many  times  ^  of  16  ? 

31.  i  of  96  is  how  many  times  |  of  12  ? 


FRACTIONS.  49 

SECTION  23.— 1.  6  is  |  of  \vhat  number  ? 

MODEL.  If  6  is  one  third  of  the  required  number,  three  thirds, 
or  the  whole,  must  be  three  times  6,  or  18.  Am.  6  is  ^  of  18. 

2.  4  is  |  of  what  number  ?      — - 

3.  7  is  TV  of  what  number  ? 

4.  9  is  ^  of  what  number  ? 

5.  3  times  4  is  |  of  what  number  ? 

6.  The  sum  of  5  and  9  is  J  of  what  number  ? 

7.  2  x  8  is  I  of  what  number  ? 

8.  |  of  21  is  |  of  what  number?    - 

9.  f  of  21  is  I  of  what  number?    ~ 

10.  ^  of  25  is  TV  of  what  number  ? 

11.  |  of  25  is  1  of  what  number? 

12.  -f  of  45  is  -J-  of  what  number  ? 

13.  2  +  8  —  3  is  £  of  what  number? 

14.  10  is  T1T  of  what  number  ?     10  is  T^  of  what? 

15.  If  a  man  can  do  -^  of  a  piece  of  work  in  2 
hours,  how  long  will  it  take  him  to  do  the  whole  ? 

MODEL.  If  a  man  can  do  one  tenth  of  a  piece  of  work  in  2 
hours,  to  do  ten  tenths,  or  the  whole,  will  require  10  times  2  hours, 
or  20  hours.  Ans.  20  hours. 

16.  If  a  family  consume  TV  of  a  barrel  of  flour  in 
11  days,  how  long  will  the  barrel  last  them  ? 

17.  If  a  boat  is  3  hours  in  performing  \  of  its  trip, 
how  long,  at  that  rate,  will  its  whole  trip  take  ? 

18.  If  a  locomotive  can  go  six  miles  in  ^  of  an 
hour,  how  many  miles  can  it  go  in  one  hour? 

19.  With  $24  collected  from  one  customer,  $16 
from  another,  and  $11  from  a  third,  a  person  paid  ^ 
of  his  taxes  ;  what  did  his  taxes  amount  to  ? 

20.  7  +  8  +  5  +  4  is  £  of  what  number  ? 

5 


r  -/r- 

50  FRACTIONS. 

21.  6  is  f  of  what  number? 

MODEL.  If  6  is  fa>o  thirds  of  the  required  number,  one  third  of  it 
is  £  of  6,  or  3 ;  and  three  thirds,  or  the  whole,  are  3  times  3,  or  9. 
An*.  6  is  f  of  9. 

/22.  4  is  f  of  what  number  ? 
/     23.  12  is  f  of  what  number  ? 

24.  36  is  T6T  of  what  number?  - 

25.  48  is  -f  of  what  number? 

26.  27  is  -f$  of  what  number? 

27.  54  is  T9^  of  what  number? 

28.  33  is  f  of  what  number  ? 

29.  40  is  -|  of  what  number? 

rl$0.  3  times  10  is  f  of  what  number? 
31.  12  — 7  is  |  of  what  number? 
32.  |  of  72  is  f  of  what  number  ? 
33.  f  of  44  is  ft  of  what  number?    J* 

34.  |  of  105  is  |  of  what  number?   - 

35.  8  is  -f-  of  what  ?     28  is  how  many  times  4  ? 

36.  8  is  -|  of  how  many  times  4  ? 

37.  9  is  T3Q-  of  how  many  times  5  ? 

MODEL.  If  9  is  three  tenths  of  a  certain  number,  one  tenth  of  it 
is  £  of  9,  or  3 ;  and  ten  tenths,  or  the  whole,  are  10  times  3,  or  30. 
30  is  6  times  5.  Ans.  6  times. 

38.  14  is  T\-  of  how  many  times  8  ? 

39.  64  is  |  of  how  many  times  12  ?  - 

40.  \  of  36  is  f  of  how  many  times  2  ?  ^ 

41.  Charles  lost  18  marbles,  which  were  -^  of  all 
he  had.     How  many  had  he  at  first  ? 

42.  A  boy,  having  received  $5  apiece  from  6  per- 
sons, had  only  T5F  of  what  he  needed  to  buy  a  pony. 
What  was  the  price  of  the  pony  ? 


FRACTIONS.  5 1 

43.  70  is  I  of  how  many  times  5  ?  / 

44.  70  is  -J  of  how  many  times  £  of  10  ? 

45.  35  is  -|  of  how  many  times  3  ? 

46.  35  is  f  of  how  many  times  £  of  12  ? 

47.  12  is  T3¥  of  how  many  times  f  of  12  ? 
48..  16  is  -^  of  how  many  times  ^  of  18  ? 

49.  £  of  80  is  how  many  times  5  ? 

50.  J  of  80  is  how  many  times  -J-  of  30  ? 

51.  3*T  of  44  is  how  many  times  ^  of  96  ? 

52.  -/Q-  of  100  is  how  many  times  6  ? 

53.  -J  of  36  is  how  many  times  ^  of  84  ? 

54.  -|  of  49  is  how  many  times  ^  of  27  ? 

55.  A  is  80  years  old,  and  T5^-  of  his  age  is  £  of  B's ; 
Low  old  is  B  ? 

56.  From  Buffalo  to  Syracuse  is  150  miles,  and  | 
of  this   distance   is  ^  the   distance   from   Buffalo  to 
Verona;  how  far  is  it  from  Buffalo  to  Verona  ? 

57.  Wilmington  is  28  miles  from  Philadelphia,  and 
J  of  this  distance  is  -^  of  the  distance  from  Philadel- 
phia to   Baltimore.     Plow  far   is   Philadelphia  from 
Baltimore  ? 

58.  From  Boston  to  Newburyport  is  36  miles,  which 
is   8  miles  more  than  |-  the  distance  from  Boston  to 
Portsmouth.     How  far  is  Portsmouth  from  Boston  ? 

59.  A  man,  having  bought  some  corn  for  $80,  sold 
it  for  %  of  its  cost ;  did  he  gain  or  lose,  and  how  much  ? 

60.  A  man  bought  some  corn  for  $80,  which  was  -J 
of  what  he  sold  it  for ;  did  he  gain  or  lose,  and  how 
much  ? 

61.  A  firmer  had  24  cows,  and  f  as  many  sheep. 
Had  he  more  sheep  or  cows,  and  how  many  more? 


52 


FRACTIONS. 


x  0 
x  2 


SECTION    24.—  1.  What  is  meant  by  Reducing  a 
fraction  ? 

2.  If  we  divide  a  pie  into  two 

__          __  e(lual  parts,  each  part  is  called 

If    we     divide 

each  half  into  two  equal  parts,  we 
get  four  equal  parts  in  all,  and 
each  is  J  of  the  whole. 

It  is  clear  that  two  of  these  fourths  equal  one 
half,  —  or  that  f  may  be  reduced  to  |. 

3.  f  =£.  What  operation  performed  on  f    2^2  _  } 
gives  £  ?    Am.  Dividing  its  terms  by  2.  4  -*-2  ~  T 

4.  i=f  .  What  operation  performed  on  \ 
gives  f  ?     ^L?i5.  Multiplying  its  terms  by  2. 

5.  What  principle  may  we  lay  down  ?    ^4n«.  The 
value  of  a  fraction  is  not  changed  by  dividing  or  mul- 
tiplying its  terms  by  the  same  number. 

6.  When  we  divide  both  terms  of  a  fraction  by  2, 
show  why  we  do  not  change  its  value.     Am.  We  get 
only  half  as  many  parts  as  before,  but  each  part  is 
twice  as  large. 

7.  When  we  multiply  both  terms  of  a  fraction  by 
2,  show  why  we  do  not  change  its  value.     Am.  We 
get  twice  as  many  parts  as  before,  but  each  part  is 
only  half  as  large. 

8.  When  is  a  fraction  in  its  lowest  terms  ?    Ans. 
When  no  number  greater  than  1  is  exactly  contained 
in  both  terms.     \  is  in  its  lowest  terms;  f  is  not,  be- 
cause 2  is  exactly  contained  in  its  numerator  and  de- 
nominator. 

9.  Is  |  in  its  lowest  terms  ?    f  ?     f  ?     |  ? 


REDUCTION    OF    FRACTIONS.  53 

10.  How  is  a  fraction  reduced  to  its  lowest  terms? 
Ans.  B}r  dividing  its  terms  by  whatever  number  or 
numbers,  greater  than  1,  are  exactly  contained  in  both. 

11.  Reduce  /T  to  its  lowest  terms.     Ans.  ^. 

12.  Reduce  the  following  to  their  lowest  terms  : — 

f.   V*.    ,V    f    «•    if-    M-    M-    A-    M. 

13.  Reduce  y^  to  its  lowest  terms. 

MODEL.     Dividing  both  terms  by  5,  we  reduce  the  fraction  to  JJ. 
Again  dividing  both  terms  by  5,  we  get  f .     Ans.  f . 

14.  Reduce  the  following  to  their  lowest  terms:  — 

*f     tt-     «•     i¥3--     T7oV     fi-     itfr- 

15.  Reduce  f-J  to  its  lowest  terms.     |f.     f 

16.  Reduce  ff  to  its  lowest  terms,     ff.     | 

1 7.  How  many  halves  in  i£  ?     In  $£-  ?     In 

18.  How  many  thirds  in  T|  ?    In  £J  ?     In 

19.  How  many  fourths  in  3|  ?     How  many  halves  ? 

20.  How  many  fifths  in  7^0-  ?     In  3T6o  ?     In  5yV  ? 

21.  Reduce  f|  to  a  mixed  number. 

NOTE.     Always  see  that  a  fraction  occurring  in  an  answer  is  in 
its  lowest  terms. 

22.  Reduce  -\«/  to  a  mixed  number.     i££.     J-p.  /£ 

23.  How  much  is  ||  of  16? 

MODEL.     £J= }.    ^  of  16  is  4,  and  f  is  3  times  4,  or  12.    ^/w.  12. 

NOTE.     It  is  often  best  to  reduce  a  fraction  to  its  lowest  terms 
before  operating  with  it. 

> ^/v 

24.  How  much  is  |f  of  6  ?     II  °f  12  ? 

25.  How  much  is  f  f  of  24  ?     f  £  of  20  ? 

26.  10  is  f£  of  vi^at  number? 

27.  I  of  16  is  f|  of  what  number? 

28.  24  is  |4  of  how  many  times  2  ? 


54  FRACTIONS. 

SECTION  25.— 1.  What  may  we  do  to  the  terras 
of  a  fraction,  without  changing  its  value  ? 

2.  How  is  a  fraction  reduced  to  its  lowest  terms  ? 

3.  How  may  a  fraction  be  reduced  to  higher  terms? 
Ans.  By  multiplying  its  terms  by  the  same  number; 

*=i- 

4.  Explain  why,  when  we  multiply  both  terms  of 
a  fraction  by  3,  we  do  not  change  its  value. 

5.  How  many  sixths  in  f  ? 

MODEL.     In  1  there  are  6  sixths,  and  in  f  there  are  f  of  6  sixths, 
or  4  sixths.     Ans.  -J. 

6.  How  many  ninths  in  ^  ?  How  many  tenths  in  f  ? 

7.  How  many  24ths  in  %  ?  In  f  ?     In  T\  ?     In  f  ? 

8.  How  many  36ths  in  f  ?  In  |  ?     In  f  ?     In  T5^  ? 
$    9.  How  many  18ths  in  f  ?  In  f  ?     In  1^  ?     In  1 f  ? 

10.  Reduce  1^  to  twentieths.   To  30ths.   To50ths. 

11.  Reduce  3|  to  twelfths.     Reduce  2|  to  63rds. 

12.  How  many  twelfths  in  J-  ?   In  |  ?   In  |  ?   In  £  ? 

13.  What  do  you  observe  with  respect  to  the  frac- 
tions T3^,  T6£,  T2^,  and  T%  ?    ^l??,s.  They  have  a  common 
denominator  (12). 

14.  Reduce  ^  and  |  to  fractions  that  have  a  com- 
mon denominator. 

MODEL.     2x5  =  10;  the  common  denominator  is  10.     ^equals 
-ro  ;  t  equals  -,%•     -4w».  1%,  «rV 

15.  Reduce  to  fractions  with  a  common  denomina- 
£  and  |;  f  and  | ;  |  and  ^_  ;  ^  and  f. 

16.  Reduce  to  fractions  with  a  common  denomina- 
tor 2}  (that  is,  f)  and  |;  |  and  1|  (f)  ;  Ij  and  If 

17.  Reduce  to  fractions  having  a  common  denomi- 
nator f  \,  and  i ;  f,  f,  and  };  ^,4,  and  f 


REDUCTION   OP   FRACTIONS.  55 

18.  If  an  ounce  is  T^  of  a  pound,  how  many  ounces 
in  J  of  a  pound  ?     In  f-  of  a  pound  ? 

19.  One  cent  is  one  hundredth  of  a  dollar;  what 
fraction  of  a  dollar  is  10  cents?   25  cents?    75  cents? 

20.  How  many  inches,  or  36ths  of  a  yard,  in  1  of 
a  yard  ?     In  |  of  a  yard  ?     In  |  of  a  yard  ? 

21.  How  many  eighths  of  a  gallon  in  a  gallon  and 
a  half?     In  5  gallons  ?     In  quarter  of  a  gallon  ? 

22.  Reduce  J,  f ,  and  -J,  to  fractions  with  the  least 
common  denominator. 

MODEL.  As  8,  the  third  denominator,  exactly  contains  the  others 
(2  and  4),  it  is  the  least  common  denominator.  •£  equals  | ;  £  equals 

fi         AvtR     A    fi    1 
b.       -ATM*,     a,    g,    8. 

23.  Reduce  to  fractions  with  the  least  common  de- 
nominator f  and  ^V     |,  J,  and  f.     J,  TV,  and  f. 

24.  Reduce  -J,  f ,  and  |f,  to  fractions  having  the 
least  common  denominator.     Reduce  ^,  f ,  £,  and  ^. 

25.  Reduce  £,  •&,  f,  and  f ,  to  fractions  with  the 
least  common  denominator.     Reduce  T5^,  ^-,  |,  |. 

26.  Reduce  T5¥,  f,  1^  (that  is,  |),  to  fractions  with 
the  least  common  denominator.     Reduce  1J,  ^,  If. 

27.  Reduce  f  and  f  to  fractions  having  a  com- 
mon denominator. 

28.  Reduce  f  and  |  to  fractions  having  the  least 
common  denominator. 

MODEL.  2,  being  a  factor  of  both  denominators  4  and  6,  may  be 
rejected  from  their  product.  4  x  6=24.  24-f-2=12,  least  common 
den.  J  equals  -,a2- ;  J  equals  |f .  Ans.  -ff,  }$. 

29.  Reduce  %  and  -f  to  fractions  having  the  least 
common  denominator.     Reduce  ^  and  f.     Reduce  f 
and  |.     Reduce  \%  and  -f.     Reduce  -£$  arid  f. 


56  FRACTIONS. 

SECTION  26.— 1.  What  is  Addition  ?     By  what 
sign  is  it  denoted  ? 

2.  What  is  the  result  of  addition  called  ? 

3.  How  much  are  9  times  5  and  ^  of  5  ? 

4.  Ho\v  much  are  6  times  7  and  ^  of  42  ? 

5.  How  much  are  4  times  6  and  ^  of  33  ? 

6.  How  many  are  2  apples  and  3  apples  ?     2  books 
and  3  books  ?     2  ninths  and  3  ninths  ? 

7.  How  many  are  4  tops  and  5  tops  ?     4  tenths  and 
5  tenths  ?     4  elevenths  and  5  elevenths  ? 

8.  How  much  are  5  thirds  and  6  thirds  ?     f +  |  ? 

9.  How  much  are  7  sixths  and  8  sixths  ?     -J  +  f  ? 

10.  What  is  the  sum  of  -|  and  -f  ?     T\  and  ^  ? 

11.  What  is  the  sum  of  \\  and  T\  ? 

MODEL.     11  fourteenths  and  5  fourteenths  are  jf,  or  l-ft-,  equal 
to  14-.     -4ns.  ly. 

12.  What  is  the  sum  of  £  and  f  ?     Off  and  |  ? 

13.  What  is  the  sum  of  T\,  -f$,  and  T6g  ? 

14.  Add  TV  TV  and  IT-     A<™  i  I,  and  f. 

15.  How  much  is  £-f  f -f  f  ?     How  much  is  -f  -f-f  ? 

16.  How  much  is  J  +  f+J-f- J? 

17.  How  much  is  7  increased  by  ^-  +  ^  ? 

18.  How  much  is  3  added  to  the  sura  of  ^  and  ^  ? 

19.  Add  4  and  |.     Add  4  and  ^+f. 

20.  Add  4,  2,  and  the  sum  of  |  and  |. 

21.  How  much  are  4^-  and  2-|  ?     3^  and  4^  ? 

22.  How  much  are  5f  and  3|  ?     If  and  7£  ? 

23.  What  is  the  sum  of  ^  of  26  and  £  of  18  ? 

24.  What  is  the  sum  of  ^,  5^,  and  2^  ? 

MODEL,     -fa,  -,Aj-,  and  -,%,  are  |f,  equal  to  1-ft,  or  l£.     1  and  5 
and  2  are  8.     ^4ws.  8k 


ADDITION   OF   FRACTIONS.  57 

25.  How  much  is  2£  +  £-h3£  ?     IfV  +  T9o+5  ? 

26.  How  much  are  If,  £  arid  ^  of  19  ? 

27.  A  person  who  owned  T^of  a  steamboat,  bought 
from  two  other  parties  T%  and  T5¥  ;  what  part  of  the 
boat  had  he  then  ? 

28.  A  man,  having  a  farm  of  253  acres,  divides  it 
equally  among  his  5  children.     If  his  eldest  child  al- 
ready had  25|  acres,  how  much  land  has  he  now  ? 

29.  Reduce  £  and  |  to  fractions  having  a  common 
denominator. 


SECTION  27.—  1.  How  much  are  f  and  f  ? 

2.  How  much  are  f  and  f  ? 

MODEL.    }=•&;  J=|J.    A  +  j|=^,  or  1-fr     ^w.  1-ft. 

3.  What  is  the   difference  between    Examples    1 
and  2? 

4.  How  do  we  add  fractions  that  have  a  common 
denominator  ? 

5.  What  must  we  first  do,  when  they  have  not  a 
common  denominator  ? 

6.  What  is  the  sum  of  £  and  £?     Of  \  and  J? 

7.  What  is  the  sum  of  }  and  -j-  ?     Of  ^  and  1  ? 

8.  What  is  the  sum  of  J  and  \  ?     Of  J  and  |  ? 

9.  AYhat  is  the  sum  of  \  and  \  ?     Of  J-  and  £  ? 

10.  How  much  is  \  +  \  ?     How  much  is  |-f^? 

11.  How  much  is  f  +  £?     How  much  is  |-f  J? 

12.  How  much  is  f  +  f?     How  much  is  |  +  f  ? 

13.  How  much  is  f  +  f  ?     How  much  is  f  +  -J? 

14.  ^-  +  |—  how  much?    ^--f  ±=  how  much  ? 

15.  |+|  +  }=howmuch?     j  +  ^  +  .=how  much? 


58  FRACTIONS. 

16.  £  +  |-f£=howmuch?    |  +  J  +  T5_— how  much? 

17.  How  much  is  1+f +  J-?     How  much  is  f -f  11? 

18.  How  much  is  1  +  2  +  f +  T1g-?     ij  +  2^? 

1 9.  How  much  is  4f  -f  2J-  ?     How  much  is  5^  +  3^  ? 

20.  How  much  is  1&  +  7&?     343¥  +  4l  ? 

21.  How  much  are  ^  of  19  and  £  of  18  ? 

22.  How  much  are  J  of  23  and  -J-  of  42  ? 

23.  How  much  are  rV  of  20,  £  of  18,  and  |  of.  10  ? 

24.  A  market-woman  sold  -J  of  her   eggs  for  40 
cents,  T1T  of  them  for  9  cents,  and  -f  of  them  for  57 
cents.     What  part  of  her  eggs  did  she  sell,  and  for 
how  much  ? 

25.  A  can  do  £  of  a  piece  of  work  in  1  day,  B  J-, 
and  C  |.     How  much  can  all  three  do  in  a  day  ? 

26.  If  a  boy  who  had  $4^,  earned  $2f  more,  and 
had  $1-3*5-  given  him,  how  many  dollars  had  he  then  ? 

27.  A  walked  half  a  mile  in  ^  of  an  hour,  f  of  a 
mile  in  ^  of  an  hour,  and  1|  miles  in  T8T  of  an  hour. 
How  many  miles  did  he  walk,  and  how  long  did  it 
take  him  ? 

28.  If  I  buy  some  muslin  for  $3J,  lace  for  82^, 
calico  for  $2^,  and  J-  dozen  collars  for  $72%,   how 
much  change  should  I  receive  for  a  twenty-dollar  bill  ? 

29.  If  a  person  buys  a  razor  for  f  of  a  dollar  and  a 
strop  for  f  of  a  dollar,  for  how  much  must  he  sell 
them  both  in  order  to  make  half  a  dollar  ? 

30.  Helen  had  83g\,  Louise  $5£§,  and  Mary  811. 
They  put  their  money  together,  and  divided  it  equally 
among  several  poor  persons,  so  that  each  received  ^ 
of  the  sum.     How  many  poor  persons  were  there,  and 
how  much  did  each  get  ? 


SUBTRACTION    OF   FRACTIONS.  59 

SECTION  28. — 1.  What  is  Subtraction  ?    By  what 
sign  is  it  denoted  ? 

2.  What  is  the  result  of  subtraction  called  ? 

3.  How  much  is  11  times  3—-^  of  40? 

4.  How  much  is  9  times  8— f  of  36  ? 

5.  How  much  is  7  times  4— f  of  twice  8  ? 

6.  4  pins— 3  pins =how  many  pins?     4  ninths— 3 
ninths = how  many  ninths?     $••—•$•? 

7.  10  knives— 3  knives=how  many  knives?     10 
elevenths— 3  elevenths  ?    H— ?V  ? 

8.  How  much  is  H""i^  ?     Ho w  much  i 

9.  How  much  is  f £— TV  ?     How  much  is 

10.  From  the  sum  of  -g%  and  ff  take  ¥%. 

11.  From  the  sum  of  fa  fj,  and  f$,  take  J-J-. 

12.  From  the  sum  of  -^  and  £J  take  H"~A' 

13.  -|  from  ^-  leaves  how  much  ?     -|  from  2|-  ? 

14.  |  from  Jy*.  leaves  how  much  ?     |  from  2f  ? 

15.  Reduce  f  and  f  to  fractions  having  a  common  (i 
denominator. 


SECTION  29.— 1.  From  |  take  |. 

2.  From  |  take  f. 

MODEL.    £  equals  fj;  f  equals  |J.     fj— |J=A-    ^w«-  A* 

3.  What  is  the  difference  between  Ex.  1  and  2  ? 

4.  How  do  we  subtract  one  fraction  from  another, 
when  they  have  a  common  denominator  ? 

5.  What  must  we  first  do,  when  they  have  not  a 
common  denominator? 

6.  From  £  take  J.     Take  |  from  £. 

7.  From  $  take  f     Take  |  from  J. 


60  FKACTIOXS. 

8.  How  much  is  f  — J  ?     How  much  is  J — 1  ? 

9.  How  much  is  f — f  ?     How  much  is  J— f  ? 

10.  How  much  is  ££—  y\?     How  much  is  £J— £? 

11.  Howmuchislf-£— ^?    How  much  is  2f|— |  ? 
/-   f  -  12.  How  much  is  3^— ^  ?  "How  much  is4J-|— 1? 

13.  How  much  is  j— J-  ?     How  much  is  1^— J  ? 

14.  How  much  is  |f — f  ?     How  much  is  1^—f  ? 

15.  How  much  is  2|—  |  ?     How  much  is  5f — ±±  ? 

16.  How  much  is  4|— J-|  ?     How  much  is  3J— |? 

17.  How  much  is  6J— f  ?     How  much  is  2£  —  J  ? 

18.  How  much  is  |—fr?     2|— J  ?     2f-li  ? 

19.  How  much  is  A-t?     4rV-|?     4A~3J? 

20.  How  much  is  5|—|  ?     4^-f  ?     3^-TV  ? 

21.  How  much  is  3f- 1£?     5^-2^?     4|— ^  ? 

22.  What  remains,  if  we  take  ^  from  5  ? 

NOTE.  We  reduce  one  of  the  5  units  to  thirteenths,  and  then 
subtract.  5=4  + 1,  or  4}f .  -ft  from  4||  leaves  4 |J.  .4^.  4H- 

23.  From  3  take  J.     Take  f  from  7.     $•  from  9. 

24.  From  5  take  f.     Take  f  from  6.     f  from  1. 

25.  From  8  take  f.     Take  ^  from  7.     ^  from  1. 

26.  From  6£  take  If. 

XOTE.  ±  =  J0 r;  4=H-  As  we  can  not  take  in  fr°m  ?V,  we  re- 
duce one  of  the  6  units  to  f  R,  and  add  it  to  -2A^  making  f  J.  l^g 
from  5f  *  leaves  4-^0-.  Ans.  4-29(J. 

27.  From  2J  take  If.     Subtract  3f  from  5f. 

28.  From  3|  take  If.     Subtract  4^  from  9f 

29.  From  9£  take  2|.     Subtract  5f  from  20. 

30.  From  50  take  40^.     Take  99^  from  100. 

31.  From  the  sum  of  \  and  T5¥  take  f. 

32.  From  £  of  28  take  the  sum  of  1J  and  f. 


SUBTRACTION    OF   FRACTIONS.  61 

SECTION  30.--1.  A  grocer,  having  a  bushel  of  po- 
tatoes, sold  J  of  it  to  one  customer,  £  to  another,  and 
T3¥  to  a  third.  What  part  remained  unsold  ? 

2.  From  a  farm  of  100  acres  were  taken  three 
fields,  containing  2^-,  4^-,  and  3T^-  acres.     Plow  many 
acres  were  left  ? 

3.  If  the  sum  of  ^  and  T\  of  a  person's  age  is  14 
years,  how  old  is  he  ? 

4.  From  two  remnants  of  calico,  containing  respect- 
ively 4f  and  8J  yards,  were  cut  12fi-  yards  for  a  dress. 
How  much  did  what  was  left  lack  of  1  yard  ? 

5.  The  difference  between  f-  and  J  of  F's  age  was 
26  years  ;  how  old  was  he  ? 

6.  A  stage,  after  making  £  and  -f  of  its  trip,  had  7 
miles  yet  to  go  ;  how  long  was  its  trip  ? 


MODEL.  i=2%;  l=28o;  2~o  +  2%=io.  The  whole  trip  was  f$ 
of  itself;  when  the  stage  had  made  |£  of  the  trip,  there  remained 
IS  —  iu>  °r  JMT-  1^  2^0"  of  the  trip  was  7  miles,  -£$  was  4-  of  7  miles, 
or  1  mile  ;  and  f  g,  or  the  whole  trip,  was  20  times  1  mile,  or  20 
miles.  Ans.  20  miles. 

7.  A  man  performed  \  of  his  journey  in  the  morn- 
^ng>  i  °f  ft  in  tne  afternoon,  and  the  rest  (12  miles)  in 
the  evening.     How  long  was  the  journey  ? 

8.  A  person  divided  $101  equally  between  his  two 
daughters.     The  elder  then  spent  $25  J,  and  gave  away 
$15|  ;  how  much  of  her  share  had  she  left? 

98  Two  thirds  of  a  certain  rod  is  blue,  f  of  it  red, 
and  the  rest  white.  If  the  white  part  is  3  inches  long, 
how  long  is  the  whole  rod  ? 

10.  How  old  am  I,  if  the  difference  between  \  and 
\  of  my  age  is  3  years  ? 


62  FRACTIONS. 

11.  Bought    some    paper  for  $12-^;    sold  it  for 
$13¥V-     By  how  much  did  the  cost  exceed  the  profit  ? 

12.  Three  pans  contained  respectively  2J,  4-|,  and 
3}  quarts  of  milk.      If  1£  quarts   were  spilled,  how 
many  quarts  were  left  ? 

13.  John  and  Cyrus  had  6  dozen  eggs  each ;  John 
sold  2£  dozen,  Cyrus  3£  dozen.     How  many  dozen  did 
Cyrus  have  left,  and  how  many  eggs  less  than  John  ? 

14.  A  can  do  a  piece  of  work  in  12  days,  and  B  in 
8  days  ;  what  part  can  each  do  in  one  day  ?  How  much 
can  both  together  do  in  one  clay  ?     After  they  have 
worked  one  day,  how  much  of  the  job  will  remain  ? 

15.  C  can  do  a  piece  of  work  in  4  days,  and  D  in 
6  days.     How  much  can  both,  working  together,  do  in 
one  day,  and  how  much  will  then  remain  to  be  done  ? 

16.  E  can  do  a  certain  job  in  9  hours,  and  F  in  6 
hours.     After  they  have  worked  together  at  it  one 
hour,  what  part  of  the  job  remains  to  be  done  ? 


SECTION  31.— 1.   What  is  Multiplication?      By 
what  sign  is  it  denoted  ? 

2.  What  is  the  result  of  multiplication  called  ? 

3.  6  times  3  equals  what  ?     5  times  3  apples  equal 
what  ?     5  times  3  sevenths  ?  .  5  times  3  ninths  ? 

4.  How  much  is  5  times  f  ?  *     5  times  f  ? 

5.  How  much  is  4  times  \  ?     6  times  f  ? 

6.  How  much  is  7  times  f  ?     3  times  f  ? 

7.  How  much  is  9  times  f  ?     8  times  f  ? 

*  Reduce  the  products  to  whole  or  mixed  numbers,  and  fractions  in  the 
answers  to  their  lowest  terms. 


MULTIPLICATION    OF   FRACTIONS.  63 

8.  Multiply  f  by  2. 

NOTE.  To  multiply  J  by  2,  we  may  double  the  number  of  parts. 
Twice  three  fourths  is  six  fourths.  Ans.  f,  or  f . 

Or,  we  may  double  the  size  of  the  parts.  Halves  are  twice  as 
great  as  fourths  (as  we  found  on  p.  52).  Hence,  twice  three  fourtfix 
is  three  halves.  Ans.  f .  The  answers  agree. 

In  the  first  case,  we  multiply  the  numerator  by  2 :       jx2=f. 

In  the  second  case,  we  divide  the  denominator  by  2  :  J+8=|. 

The  second  method  is  shorter,  when  it  can  be  used,  because  it 
brings  the  answer  in  its  lowest  terms  at  once. 

9.  How  much  is  -^  x  5  ?     ^  x  2  ?     f  x  3  ? 

10.  Multiply  -j5g-  by  4.     ^  by  8.     -^  by  9. 

11.  Multiply  ^  by  3.     Ji  by  10.     $£  by  8. 

12.  What  is  the  product  of  ¥\  and  6  ?     ^and  12  ? 

13.  What  is  the  product  of  fa  and  5  ?    -^  and  11  ? 

14.  What  is  the  product  of  ^f  and  9  ?     f|  and  8  ? 

15.  What  is  the  product  of  10  and  f  ?    T4¥  and  7  ? 

16.  What  two  ways  are  there  of  multiplying  a  frac- 
tion by  a  whole  number  ? 

1 7.  Which  is  preferable  ?     Why  ? 

18.  What  is  4  times  |- ?     4  times  f?     4  times  f? 

19.  When  you  multiply  a  fraction  by  its  own  de- 
nominator, what  do  you  get  ? 

20.  How  much  is  5  times  f  ?     10  times  T7¥? 

21.  How  much  is  9  times  f  ?     19  times  -f4  ? 

22.  How  much  is  100  times  yfg- ?     25  times  f|? 

23.  How  much  does  4  times  fa  lack  of  10  ? 

24.  What  must  be  added  to  15  times  ^f ,  to  make  7  ? 

25.  Which  is  greater,  5  times  T2T  or  2  — 1|? 

26.  7  times  ^  is  |-  of  what  number?  • 

27.  8  times  -^  is  £  of  what  number  ?  - 

28.  3  times  -^T  is  f  of  what  number? 


64  FRACTIONS. 

SECTION  32.— 1.  How  much  is  5  times  3-|  ? 
MODEL.     5  times  f  is  3£,  or  2£ ;  5  times  3  is  15.     15  +  2?=17f-. 
/4?w.  17f. 

2.  How  much  is  6  times  2J?     4  times  8f  ? 

3.  How  much  is  3  times  5| ?     7  times  1^  ? 

4.  How  much  is  9  times  4-J  ?     5  times  2£J  ? 

5.  How  much  is  10  times  7|-  ?     12  times  3f  ? 

6.  Multiply  6f  by  2.     By  5.     By  7.     By  10. 

7.  What  cost  8  vests,  at  $5f  each  ? 

8.  How  much  ale  in  6  cans,  holding  2f  pints  each  ? 

9.  What  will  9  hens  weigh,  averaging  2|-  pounds  ? 

10.  Multiply  6  by  J. 

NOTE.  6  multiplied  by  %  must  be  *  as  much  as  6  multiplied  by 
1, — that  is,  J  of  6.  Multiplying  a  number  by  *,  i,  &c.,  is  therefore 
equivalent  to  taking  *,  £,  &c.,  of  that  number. 

11.  Multiply  24  by  J.     By  f.     By  1      By  f. 

12.  Multiply  18  by  f     By  f.     By  f.     By  f. 

13.  How  much  is  £  of  2  ? 

MODEL.     J  of  1  is  J,  and  J  of  2  is  twice  £,  or  f.     -4ws.  f . 

14.  How  much  is  \  of  2?     iof2?     TVof7? 

15.  How  much  is  }  of  3  ?     J  of  5  ?     TV  of  6  ? 

16.  Multiply  6  by  ^.     Multiply  4  by  TV     7  by  /f. 

17.  How  much  is  f  of  21  ? 

MODEL.  One  sixth  of  21  is  3$,  and  five  sixths  are  5  times  3£,  or 
17*.  ^ws.  17*. 

18.  How  much  is  f  of  5?     |  of  9  ?     f  of  11  ? 

19.  How  much  is  f  of  8  ?     f  of  9  ?     •&  of  14  ? 

20.  How  much  is  £  of  2  ?     f  of  2  ?     $  of  2  ? 

21.  How  much  is  J  of  3  ?     ^of4?     -^oflG? 

22.  How  much  is  £  of  100  ?     f  of  100  ?     f  of  100  ? 

23.  How  much  is  7  times  2f-  ?    f  of  4  ?     -f  of  11  ? 


MULTIPLICATION    OF   FRACTIONS.  65 

24.  How  far  will  a  locomotive,  moving  at  the  rate 
of  20T^  miles  an  hour,  go  in  5  hours  ?     In  8  hours  ? 

25.  What  cost  two  pieces  of  meat,  weighing  re- 
spectively 8-|  and  6^-g-  pounds,  at  20^  cents  a  pound  ? 

26.  At  $8f  an  acre,  what  will  be  the  cost  of  3 
fields  containing  3f  acres  each  ? 

27.  What  is  the  weight  of  four  cheeses,  if  two 
them  weigh  9f  pounds  each,  and  the  other  two 
pounds  each  ? 

28.  If  I  divide  $2  equally  among  8  beggars,  what 
part  of  $2  will  each  receive  ?     What  part  of  $1  ? 

29.  A  person  bought  56  pounds  of  butter,  and  sold 
f  of  it ;  how  many  pounds  remained  on  hand  ? 

30.  Emma  is  3|  years  old  ;  Laura  is  5  times  as  old 
as  Emma.     In  how  many  years  will  Laura  be  21  ? 


SECTION  33,— 1.  How  much  is  5f  times  7? 

MODEL.     5  times  7  is  35 ;  f  of  7  is  V~>  or  4|;    35  +  4|=39|. 
Ans.  39J. 

2.  How  much  is  2£  times  24  ?     3f  times  14  ? 

3.  How  much  is  5f  times  10  ?     4f  times  15  ? 

4.  How  much  is  8f  times  12?     6|  times  16  ? 

5.  How  much  is  3J  times  $  of  18  ? 

6.  What  cost  ?!  acres,  at  $10  an  acre? 

7.  If  f  of  a  ton  of  hay  cost  $15,  what  will  1  ton 
cost  ?    What  will  £  of  a  ton  cost  ? 

8.  If  |  of  a  ton  of  hay  cost  $20,  what  will  f  of  a 
ton  cost  ?     What  will  2f  tons  cost  ? 

9.  What  are  3^  gallons  of  petroleum  worth,  if  | 
of  a  gallon  is  worth  21  cents  ? 


60  FRACTIONS. 

10.  |-  of  4  equals  how  many?     -|-  of  4  apples  ?    £ 
of  4  ninths?     Jof^?    Jof-^? 

11.  ^  of  6  equals  how  many?     ^  of  6  knives  ?     ^ 
of  6  sevenths  ?    £  of  £  ?     J-  of  ^  ? 

12.  How  much  is  J  of  f  ?     £  of  f J  ?    TV  of  f°-? 


13.  What  is  multiplying  by  a  fraction  equivalent 
to  ?     Ans.  To  taking  such  a  part  as  is  denoted  by  the 
fraction.     Multiplying  by  %  is  equivalent  to  taking  -J-. 

14.  Multiply  |f  by  f     Multiply  ff  by  TV     Multi- 
ply ^  by  |.     Multiply  -|f  by  ^. 

15.  How  much  is  f  of  *£-  ? 

MODEL.     One  fourth  of  Lf-  is  f ,  and  three  fourths  are  3  times  ^ 
or  ^, — equal  to  1^.     Ans.  1J. 

16.  How  much  is  f  of  |-|?     How  much  is  f  of  ff-? 

17.  How  much  is  J  of  fj  ?  How  much  is  f  of  1||  ? 

18.  How  much  is  f  of-2/  ?     How  much  is  f  of  2f  ? 

19.  How  much  is  f  of  -^  ?    How  much  is  £  of  1^  ? 

20.  How  much  is  £  of  f  of  -^-? 

*  A       21.  How  much  is  f  of  If  ?    How  much  is  f  of  f  ? 

22.  How  much  is  f  of  f  of  If  ? 

23.  How  much  is  f  of  f  of  2f|? 

24.  If  a  person  bought  f  of  a  ship,  and  then  sold 
£  of  his  share,  what  part  of  the  ship  did  he  retain  ? 

25.  When  cheese  is  -f^  of  a  dollar  a  pound,  how 
much  will  f  of  a  pound  cost  ? 

26.  Jane's  age  is  J  of  Clara's,  and  Clara's  is  f  of 
Lucy's.     If  Lucy  is  18,  how  many  years  old  is  Jane  ? 

27.  If  a  cow  eats  -fa  of  a  ton  of  hay  in  a  week, 
what  will  her  daily  supply  of  hay  cost  when  hay  is 
$21  a  ton? 

4/s 


DIVISION    OF   FRACTIONS.  '  67 

SECTION  34.— 1.    What   is  Division  ?    By  what 
sign  is  it  denoted  ? 

2.  What  is  the  result  of  division  called  ? 

3.  Dividing  by  2  is  equivalent  to  taking  what  part  ? 

4.  Dividing  by  3  is  equivalent  to  what  ?     Dividing 
by  4?     Dividing  by  7  ? 

5.  How  many  times  is  6  contained  in  18  ?     How 
much  is  i  of  18  ?     Divide  18  by  6. 

6.  How  many  times  is  6  contained  in  18  sevenths  ? 
How  much  is  %  of  if-  ?     Divide  if-  by  6. 

7.  Divide  |  by  2. 

NOTE.     To  divide  f  by  2,  we  may  take  £  of  the  number  of 

parts  ;  £  of  eight  halves  is  four  halves,  or  2.     Ans.  2. 

Or,  we  may  make  each  part  half  as  great.     FourtJis  are  half  as 

great  as  halves  (as  we  found  on  p.  52).     Hence,  £  of  8  halves  is  8 

fourths,  or  2.     Ans.  2.     The  answers  agree. 

In  the  first  case,  we  divide  the  numerator  by  2  :  £**=£= 2 
In  the  second  case,  we  multiply  the  denominator  by  2  :  £x2=i  —  2 
The  first  method  is  preferable  when  the  whole  number  is  exactly 

contained  in  the  numerator  of  the  fraction. 

8.  What  two  ways  are  there  of  dividing  a  fraction 
by  a  whole  number?     Which  is  preferable? 

9.  When  can  the  first  method  be  used  ? 

10.  How  many  times  is  6  contained  in  f  ? 

NOTE.    As  6  is  not  exactly  contained  in  2,  we  have  to  use  the 
second  method:  f  x  6—-&i  or  JM-     ^ns-  ~2T- 

11.  Divide  f  by  9.     Divide  f  by  6.     Divide  f  by  8. 

12.  How  many  times  is  10  contained  in  T9T  ?  In  -i~|  ? 

13.  How  much  is  J  of  f  ?    $  of  TV?     -J-  of  •&  ? 
'    14.  How  much  is  f  of  f  ? 

MODEL.     One  fourth  of  f  is  /0-,  or  -fa ;  and  three  fourths  are  3 
tim«s  -fa,  or  -fc     Ans.  -,30. 


08  FRACTIONS. 

15.  How  much  is  £  of  f  ?    £  of  f  ?     f  of  f  ? 

16.  How  much  is  £  of  if  ?     i  of  If  ?     f  of  If  ? 

17.  How  much  is  |  of  f  ?    1  of  1£  ?    f  of  1|  ? 

18.  How  much  is  f  of  2|  ?     f  of  2|  ?     f  of  3|  ? 

19.  A,  B,  and  C,  own  a  ferry,  having  equal  shares. 
B  sells  f  of  his  share  to  C ;  what  part  of  the  ferry  does 
B  own  after  this  sale,  and  what  part  C  ?    How  much 
more  has  C  than  B  ? 

20.  If  1  pound  of  butter  costs  f  of  a  dollar,  how 
much  will  J  of  a  pound  cost  ? 

21.  Mary  has  $1-^  and  George  $2f.     They  divide 
•J  of  what  they  both  have  equally  between  two  poor 
persons.     What  part  of  a  dollar  does  each  receive  ? 

22.  By  how  much  does  f  of  T^  exceed  \  of  f-  ? 

23.  How  much  is  3|  times  2J  ? 

MODEL.     2}=-^.    3  times  *£=*£,  or  8 J.     One  third  of  ^  is  f^ 
and  two  thirds  are  twice  -J-J,  or  \* =lf.    8£+  1  J  =  10f2-.    Am.  10 -,V 

24.  How  much  is  2J-  times  1 J  ?     If  times  4£  ? 

25.  How  much  is  4^  times  60J?     3|  times  41|? 

26.  How  much  is  5|  times  10TV?     6£  times  18TV? 


SECTION  35. — 1.  How  many  halves  are  there  in  1? 
In  2  ?     In  5  ? 

2.  How  many  times  is  £  contained  in  1  ?     In  2  ? 

3.  Dividing  by  ^  is  equivalent   to  what  ?    Ans. 
Dividing  by  1  is  multiplying  by  2. 

4.  Multiplying  by  ^  is  equivalent  to  what  ?     Ans. 
Multiplying  by  \  is  taking  ^,  or  dividing  by  2. 

5.  If  we  divide  8  by  -J,  what  is  the  quotient  ?     If 
we  multiply  8  by  £,  what  is  the  product  ? 


DIVISION    OF   FRACTIONS.  69 

6.  Dividing  by  -^  is  equivalent  to   what?     Am. 
Dividing  by  |  is  multiplying  by  3. 

7.  Dividing  by  4-  is  equivalent  to  what  ?     By  \  ? 

8.  How  many  times  is  ^  contained  'in  3  ?     In  9  ? 

9.  How  many  times  is  ^  contained  in  2  ?     In  3-^-  ? 
In4TV?     In6f?     In7|? 

10.  How  many  times  £  is  3|  ?     4-£-  ?     5f  ?     6|  ? 

11.  How  many  times  %  is  f  ?     Divide  f  by  £. 

12.  How  many  times  \  is  |-  ?    Divide  f  by  ^-. 

13.  How  many  times  ^  is  -f-  ? 

14.  How  many  times  T^  is  If? 

15.  How  many  times  is  -^  contained  in  1^  ? 

16.  How  many  times  is  -^  contained  in  2^  ? 

17.  How  many  times  is  $  contained  in  f  ? 

MODEL.  One  sixth  is  contained  in  f ,  Lf  or  f  times ;  and  five 
sixths,  being  5  times  as  great,  is  contained  in  it  i  of  J  times,  or  -,90- 
times.  Ans.  <$,-  times. 

18.  How  many  times  is  f  contained  in  f  ?     f  in  f  ? 

19.  How  many  times  is  f  contained  in  f  ?     |  in  -|? 

20.  f  in  ^-  how  many  times?     -|  in  If?         ^ 

21.  -|  in  ^  how  many  times  ?     f  in  1^-  ? 

22.  ^  in  4|  how  many  times  ?     -^  in  1^? 

23.  f  in  f  how  many  times?     If  in  f  ? 

24.  J  in  |  how  many  times  ?     If  in  1£? 

25.  if  in  2_2  now  mariy  times?     If  in  34-? 

26.  3-^  in  2f  how  many  times  ?     4|  in  5|? 

27.  How  far  will  a  person,  walking  at  the  rate  of 
2-|  miles  an  hour,  walk  in  3^  hours  ? 

28.  If  the  current  of  a  river  moves  2f  miles  in  3J 
hours,  how  far  will  it  move  in  1  hour  ? 

29.  How  much  is  5£  x  2f  ?     How  much  is  6^-7- 2f  ? 


70  FRACTIONS. 

SECTION  36.— 1.  How  many  times  is  9  contained 
in  lOOf ? 

MODEL.     9  is  contained  in  100,  11  times  and  1  over.     9  is  con- 
tained in  If,  or  2,  i  time.     Ans.  \\\  times. 

2.  How  many  times  is  11  contained  in  84^? 

3.  How  many  times  is  7  contained  in  42|^? 

4.  How  many  times  is  12  contained  in  39f? 

5.  How  many  times  is  5  contained  in  43f  ? 

6.  How  many  times  is  4  contained  in  49f  ? 

7.  How  many  times  are  7  apples  contained  in  14 
apples  ?     7  cups  in  14  cups  ?     7  ninths  in  14  ninths  ? 

8.  How  many  times  are  5  pins  contained  in  45  pins  ? 
5  eighths  in  45  eighths  ?     -|  in  ^  ?     ^  in  *f-  ? 

9.  How  do  we  divide  one  fraction  by  another,  when 
they  have  a  common  denominator?     ^Lns.  Divide  nu- 
merator by  numerator,  rejecting  the  denominators. 

1(>.  -4<5M-f  =  now  many?      ^M-£?      If-v-f?     5-f-v- 

119       J.4    •    I  2  9       q  R     -ISO 
18'       »f"s"1f-r       t5T2-~1T3i 

11.  What  cost  |  of  a  barrel  of  cider,  at  $8£  a  bar.  ? 

12.  A  person  laid  out  $55J  for  fuel.     He  bought  5 
tons  of  coal  at  87 J  a  ton,  and  spent  the  rest  for  wood 
at  $3 1-  a  load  ;  how  many  loads  of  wood  did  he  buy  ? 

13.  If  a  horse  goes  2^  miles  in  an  hour,  what  part 
of  an  hour  will  it  take  him  to  go  one  mile  ? 

14.  A  farmer  sold  f  of  his  flock,  and  had  17  sheep 
left;  how  many  sheep  had  he  originally? 

15.  A  farmer  sold  14  sheep,  and  had  f  of  his  flock 
left ;  how  many  sheep  had  he  left  ? 

16.  What  will  a  pile  of  16f  bushels  of  potatoes 
bring,  if  put  up  in  bags  holding  2|  bushels  each,  and 
sold  for  8lTV  per  bag  ? 


FRACTIONS.  71 

17.  7  is  f  of  what  number  ? 

MODEL.  If  7  is  two  thirds  of  a  certain  number,  one  third  of  it  is 
£  of  7,  or  I ;  and  three  thirds,  or  the  whole,  are  3  times  J,  or  ^ 
which  equals  10£.  Ans.  10£. 

18.  9  is  f  of  what  number  ? 

19.  12  is  f  of  what  number  ? 

20.  |  of  20  is  |  of  what  number? 

21.  ^  of  30  is  |  of  how  many  times  8  ? 

22.  -J  of  32  is  |  of  how  many  times  |? 

23.  £  of  16  is  |  of  how  many  times  10  ? 

24.  f  of  25  is  f  of  how  many  times  12  ? 

25.  f  of  63  is  -J-  of  how  many  times  J  of  24  ? 

26.  |  of  40  is  |  of  how  many  times  3|  ? 

27.  |  is  f  of  what  number? 

28.  |  is  f  of  what  number? 

29.  |f  is  |  of  what  number  ? 

30.  f  is  f  of  what  number? 

31.  If  is  |f  of  what  number? 

32.  If  is  |  of  what  number  ? 

33.  f  is  |^  of  how  many  times  f? 

MODEL.  If  f  is  four  thirds  of  a  certain  number,  one  third  of  it 
is  £  of  f ,  or  -23y ;  and  three  thirds,  or  the  whole,  are  3  times  -/-$,  or 
ro-  29o  is  as  many  times  f  as  f  is  contained  times  in  f0.  One 
sixth  is  contained  in  2%  6  times  2%,  or  f £,  which  equals  f },  times ; 
and  Jive  sixths  is  contained  in  it  -t  of  f£  times,  or  f  £.  ^?w.  f  J. 

34.  |-|  is  f  of  how  many  times  1£  ? 

35.  £  of  li  is  |  of  how  many  times  1|  ? 

36.  |  of  -fif  is  4  times  what  number  ? 

37.  f  of  l^j  is  |-  of  |  of  what  number  ? 

38.  |  of  24-  is  |  of  J£  of  how  many  times  2  ? 

39.  |  of  3f  is  Y\F  of  how  many  times  |  ? 


72  FEDERAL   MONEY. 

40.  If  a  grocer  buys  cheese  for  14|  cents  a  pound, 
and  sells  it  for  16  cents,  how  many  pounds  will  he 
have  to  sell  in  order  to  make  7-J  cents  ? 

41.  Suppose  that  it  takes  2-J  yards  of  merino  to 
make  a  sack,  and  7J  yards  to  make  a  dress.     After  3 
sacks  have  been  cut  from  a  piece  containing  22  f  yards, 
how  many  dresses  can  be  cut  from  what  remains  ? 


CHAPTER       SIXTH. 

FEDERAL    MONEY. 

SECTION  37. — 1.    One  hundred  cents  make  $1 ; 
how  many  cents  are  there  in  $5  ? 

MODEL.     Since  there  are  100  cents  in  $1,  in  $5  there  are  5  times 
100  cents,  or  500  cents.     Ans.  500  cents. 

2.  One  hundred  cents  make  $1 ;  how  many  dollars 
in  500  cents  ? 

MODEL.     Since  there  are  100  cents  in  $1,  in  500  cents  there  are 
as  many  dollars  as  100  cents  are  contained  times  in  500  cents,  or  5. 

Ans.  $5. 

3.  When  we  change  $5  to  500  cents,  do  we  alter 
the  value  ?     Do  we  alter  the  form  ?    What  is  this  pro- 
cess called  ?    Ans.  Reduction. 

4.  When  we  change  dollars  to  cents,  do  we  go  to 
a  higher  or  lower  denomination  ?     What  is  this  kind 
of  Reduction  called  ?    Ans.  Reduction  Descending. 

5.  When  we  change  cents  to  dollars,  do  we  go  to 
a  higher  or  lower  denomination  ?     What  is  this  kind 
of  Reduction  called  ?    Ans.  Reduction  Ascending. 


REDUCTION    OF    FEDERAL   MONEY.  73 

SECTION  38.—  What  is  Federal  Money?     Ans. 
Federal  Money  is  the  currency  of  the  United  States. 

TABLE  OF  FEDERAL  MONEY. 

10  mills  (in.)  make  1  cent,   .  .     .     .     c.,  ct. 
10  cents,                    1  dime,    .     .     .     .  di. 

10  dimes,  1  dollar,  .     .     .     $. 

10  dollars,  1  eagle,  .     .     .     .  E. 

Accounts  are  kept  in  dollars  and  cents.     Cents  are  written  at 
the  right  of  dollars,  with  a  period  between,  and  occupy  two  places. 
The  first  place  is  filled  with  a  naught,  if  the  cents  are  expressed  by 
but  one  figure.     Mills  are  written  at  the  right  of  cents.     Thus  :  — 
Nine  dollars,  5  cents,   .....         $9.05 
Nine  dollars,  fifty  cents,  .....     $9.50 
Nine  dollars,  fifty  cents,  five  mills,       .        .        $9.505 
Nine  dollars,  five  mills,    .....     $9.005 
As  100  cents  make  a  dollar,  1  cent  is  j^  of  a  dollar,  2  cents 
T^O>  &c-     Cents  are  sometimes  written  as  hundredths  of  a  dollar  ; 
five  dollars  and  twenty-five  cents  may  be  written 


1.  How  many  mills  in  3  cents  ?  *  In  7c.  ?  In  19c.  ? 

2.  How  many  cents  in  5  dimes  ?     In  11  dimes? 

3.  How  many  dimes  in  $2  ?     In  $8  ?     In  $10  ? 

4.  How  many  dollars  in  4  eagles?    In  12  eagles  ? 

5.  How  many  dollars  in  50  dimes?*   In  170  dimes? 

6.  How  many  eagles  in  $60  ?     In  $90  ?     In  $20  ? 

7.  How  many  dimes  in  50  cents  ?   In  90c.  ?  In40c.  ? 

8.  How  many  cents  in  $5  ? 

MODEL.  Since  10  cents  make  1  dime,  and  10  dimes  make  $1,  in 
$1  there  must  be  10  times  10,  or  100,  cents  ;  and  in  $5,  5  times  100, 
or  500,  cents.  Am.  500  cents. 

*  See  MODELS,  on  page  72. 


74  FEDERAL   MONEY. 

9.  How  many  cents  in  $9  ?  In  4  eagles  ?  How  many 
dimes  in  7  eagles  ?   How  many  mills  in  3  dimes  ?  In  $2  ? 

10.  How  many  cents  in  6  dollars  35  cents  ? 

MODEL.  In  $1  there  are  100  cents,  and  in  $6  six  times  100  cents, 
or  600  cents.  600  cents  +  35  cents =635  cents.  Ans.  635c. 

In  such  cases,  to  reduce  to  cents  we  need  only  remove  the  pe- 
riod and  dollar-mark : — 

$6.35=635  cents 

11.  How  many  cents  in   $4.89?     In  $10.10?     In 
~$9.05  ?     How  many  mills  in  $2.375  ?     In  87c.  5m.  ? 

12.  How  many  dollars  in  635  cents  ? 

MODEL.  100  cents  make  $1,  and  635  cents  will  make  as  many 
dollars  as  100  cents  are  contained  times  in  635  cents,  or  6-^fj.  Ans. 
:£6-ft&,  or  $6.35. 

In  such  cases,  we  need  only  cut  off  the  two  right-hand  figures 
for  cents,  and  what  remains  on  the  left  will  be  dollars : — 
635  cent3=,$6.35 

13.  How  many  dollars  in  757  cents?     In  843  cents? 
In  926  cents  ?     In  4270  mills? 

14.  How  many  eagles  in  $17  ?     How  many  cents  in 
53  mills  ?     How  many  dimes  in  47  cents? 

15.  How  many  cents  in  half  a  dime?  In  1 9|- dimes  ? 
In  5|  dollars  ?     How  many  dollars  in  5  half-eagles  ? 

16.  How  many  cents  in  half  a  dollar?     In  \  of  a 
-dollar?     In  £  of  81?     In  $t  ?     In  $£?     In  $-1  ? 

17.  What  part  of  a  dollar  is  50  cents  ? 

MODEL.  Since  100  cents  make  a  dollar,  1  cent  is  TOTT  of  $1, 
;  and  50  cents  are  50  times  Tfoj,  or  -&%  which  equals  £.  Am.  i  of 
.-a  dollar. 

18.  What  part  of  a  dime  is  3  cents  ?    What  part  of 
:an  eagle  is  $4  ?     What  part  of  a  cent  is  6  mills  ? 


REDUCTION.  75 

19.  What  part  of  a  dollar  is  75  cents?     33^  cents? 
10  cents?    12J  cents?   25  cents  ?   20 cents?   16|  cents? 

20.  What  cost  44  Grammars,  at  75c.  apiece  ? 
MODEL.     75c.  is  f  of  $1.     44  Grammars,  at  $1  apiece,  would 

cost  $44,  and  at  f  of  a  dollar,  they  cost  J  of  $44,  or  $33.     Ans.  $33. 

21.  What  cost  66  rulers,  at  12|-  cents  apiece  ? 

22.  What  cost  2  dozen  Readers,  at  33-|c.  each  ? 

23.  At  25  cents  each,  what  cost  18  slates? 

24.  At  16f  cents  each,  what  cost  40  magazines  ? 

25.  At  50c.  each,  how  many  books  can  I  buy  for  $5  ? 

26.  At  33^c.  each,  how  many  balls  can  I  buy  for  $9  ? 


CHAPTER      SEVENTH. 

REDUCTION. 

SECTION  39,— 1.  What  is  Reduction  ?  Ans.  Re- 
duction  is  changing  a  quantity  from  one  denomina- 
tion to  another,  without  altering  its  value. 

2.  What  is  Reduction  Ascending  ?     Ans.  Reduc- 
tion Ascending  is  changing  a  quantity  from  a  lower 
denomination  to  a  higher,  without  altering  its  value. 

3.  What  operation  do  we  use  in  Reduction  Ascend- 
ing ?    Ans.  Division. 

4.  What  is  Reduction  Descending  ?    Ans.  Reduc- 
tion Descending  is  changing  a  quantity  from  a  higher 
denomination  to  a  lower,  without  altering  its  value. 

5.  What  operation  do  we  use  in  Reduction  De- 
scending?   Ans.  Multiplication. 


76  REDUCTION. 

SECTION  40.— What  is  English  or  Sterling  Money  ? 
Ans.  Tb^  currency  of  Great  Britain. 

TABLE  OF  STEELING  MONEY. 

4  farthings  (far.,  qr.),  1  penny,  .     .     d. 

12  pence,  1  shilling, .     .  s. 

20  shillings,  1  pound, .     .     £. 

21  shillings,  1  guinea,  .     .  guin. 

1.  How  many  pence  in  £1  5s.  6d.  ? 

MODEL.  In  £1  are  20s.  20s.  +  5s.=25s.  In  Is.  are  12d.,  and 
in  25s.  25  times  12d.,  or  300d.  300d.  +  6d.=306d.  Am.  306d. 

2.  How  many  pounds,  &c.,  in  306  pence  ? 

MODEL.  12d.  make  Is. ;  306d.  will  therefore  make  as  many 
shillings  as  12  is  contained  times  in  306,  or  25s.  and  6d.  over.  20s. 
make  £1 ;  25s.  will  therefore  make  as  many  pounds  as  20  is  con- 
tained tunes  in  25,  or  £1  and  5s.  over.  Am.  £1  5s.  6d. 

3.  How  many  shillings  in  £9  3s.  ?    In  4  guin.  7s.  ? 

4.  How  many  farthings  in  9s.  8d.  ?    In  lid.  3qr.  ? 

5.  How  many  pounds,  &c.,  in  73s.  ?    In  251d.  ? 

6.  How  many  pence  in  £5  ?     In  £6  lid.  ? 

7.  How  many  shillings,  &c.,  in  89  far.  ?    In  5  guin.  ? 

8.  How  many  pence  in  1  guin.  4s.  ?     In  %  of  £1  ? 

9.  How  many  farthings  in  3s.  9fd.  ?     In  ll|d.  ? 

10.  How  many  pence  in  £1  ?     Farthings  in  Is.  ? 

11.  What  part  of  a  pound  is  5s.  ?     4s.  ?     10s.  ? 

12.  What  part  of  a  shilling  is  4d.  ?     8d.  ?     9d.  ? 

13.  At  6d.  a  pound,  what  cost  40  pounds  of  meat? 
NOTE.   Reduce  the  answers  to  the  highest  denomination  possible, 

14.  What  cost  8  knives,  at  3s.  each  ? 

15.  What  cost  6  dozen  slates,  at  8d.  each? 


TROY    WEIGHT.  77 

SECTION  41.— For  what  is  Troy  Weight  used  ? 
A/is.  For  weighing  gold,  silver,  and  precious  stones. 

TABLE  OF  TROY  WEIGHT. 

24  grains  (gr.)  make  1  pennyweight,   pwt. 
20  pennyweights,        1  ounce,   .     .     .  oz. 
12  ounces,  1  pound,     .     .     To. 

1.  Reduce  86  oz.  to  pounds,  &G.   To  pennyweights, 

2.  How  many  grains  in  1  oz.  ?     In  5  pwt.  20  gr.  ? 

3.  How  many  pounds  in  250  pwt.  ?     In  100  oz.  ? 

4.  How  many  grains  in  4  oz.  3  pwt.  ? 

5.  How  many  ounces  in  J  of  a  Ib.  ?     In  5-f-  Ib.  ? 

6.  How  many  pennyweights  in  \  Ib.  ?     In  4|-  oz.  ? 

7.  How  many  grains  in  f  of  a  pwt.  ?    In  2f  pwt.  ? 

8.  How  many  grains  in  \  of  an  ounce  ?  In  f  pwt.  ? 

9.  What  part  of  a  pwt.  is  18  grains  ?     14  grains? 
30.  What  part  of  an  ounce  is  15  pwt.  ?     1  grain  ? 

11.  What  part  of  an  ounce  is  -fa  of  a  pound  ? 

MODEL.     1  Ib.  is  12  oz. ;  and  -^r  of  a  Ib.  is  -^  of  12  oz.,  or  J§  of 
1  oz.     iJ=f.     ^4n*.  I  oz. 

1 2.  What  part  of  a  pennyweight  is  -fa  of  an  ounce  ? 

13.  What  part  of  a  grain  is  -fa  of  a  pennyweight?1 

14.  What  part  of  a  pound  is  120  pwt.  ? 

15.  A  jeweller  bought  7  pwt.  of  pure  gold,  9  pwt. 
of  gold  coin,  and  16  pwt.  of  silver  ;  how  many  ounces, 
did  he  buy  altogether  ? 

16.  What  is   the   weight  in  pounds,   &c.,  of  12 
spoons,  of  30  pwt.  each  ? 

17.  How  many  spoons,  weighing  35  pwt.  each,  canu 
be  made  out  of  1  Ib.  2  oz.  of  silver  ? 


78  REDUCTION. 

SECTION  42.— By  whom  is  Apothecaries'  Weight 
used  ?    Ans.  By  apothecaries,  in  mixing  medicines, 

TABLE  OF  APOTHECARIES'  WEIGHT. 

20  grains  (gr.)  make  1  scruple,  .     sc.  or  3. 

3  scruples,  1  dram,.     .  dr.  or  3. 

8  drams,  1  ounce,    .     oz.  or  3 . 

12  ounces,  1  pound,     .  Ib.  or  ft. 

1.  How  many  scruples  in  5  oz.  ?     In  1  Ib.  ? 

2.  How  many  drams  in  95  gr.  ?     In  4  Ib.  5  oz.  ? 

3.  How  many  grains  in  3  dr.  5  sc.  ?    In  J  Ib.  ? 

4.  Ho\v  many  pounds  in  113  3  ?     In  113  §  ? 

5.  Reduce  500  gr.  to  ounces,  &c.     -|  oz.  to  scruples. 

6.  What  part  of  a  scruple  is  17  gr.  ?     8  gr.  ? 

7.  What  part  of  an  ounce  is  3  drams?     2  sc.  ? 

8.  What  part  of  a  pound  is  5  oz.  ?    -J-  dram  ? 

9.  What  fraction  of  a  dram  is  -fa  of  an  ounce  ? 

10.  What  part  of  a  grain  is  -^  of  a  scruple  ? 

11.  How  much  are  f  of  a  dram  and  ^  of  an  ounce? 

MODEL.     In  1  oz.  are  8  dr.,  and  in  £  of  an  ounce  J  of  8  drams, 
or  2|  drams.     2|  drams  +  J  dr.=3 ,%  drams.     Ans.  3  ^  dr. 

12.  How  much  are  f  of  a  Ib.  and  ^  of  an  ounce  ? 

13.  How  much  are  f  of  a  dram  and  3J  sc.  ? 

14.  How  much  are  f  of  an  ounce  and  ^  of  a  dram  ? 

15.  How  many  drams,  &c.,  will  it  take  for  20  pow- 
ders, each  containing  20  grains  ? 

16.  How  many  powders,  of  1 J  sc.  each,  can  be  put 
up  from  an  ounce  of  soda  ? 

17.  If  a  druggist  charges  50c.  for  ten  powders,  con- 
taining 15  gr.  each,  at  what  rate  is  that  per  ounce? 


AVOIRDUPOIS   WEIGHT.  79 

SECTION  43.— For  what  is  Avoirdupois  Weight 
used  ?  Ans.  For  weighing  groceries,  meat,  coal,  cot- 
ton, drugs  when  sold  in  quantities,  and  all  articles 
except  gold,  silver  and  precious  stones. 

TABLE  OF  AVOIRDUPOIS  WEIGHT. 
16  drams  (dr.)  make  1  ounce,      .     .     .     oz. 
16  ounces,  1  pound, .     .     .     .  Ib. 

25  pounds,  1  quarter,  .     .     .     qr. 

4  quarters,  1  hundred- weight,  cwt. 

20  hundred-weight,     1  ton,     .     .     .     .     T. 

1.  How  many  drams  in  a  pound  ?     Ounces  in  a 
quarter?     Pounds  in  1  cwt.  ?     Pounds  in  a  ton  ? 

2.  How  many  pounds  in  5  T.  3  cwt.  21  Ib  ?     In  \ 
of  a  ton  ?     In  35  ounces  ?     In  4£  T.  ? 

3.  How  many  ounces  in  5£  Ib.  ?     In  83  dr.  ?     In  2 
cwt.  10  Ib.  5  oz.  ?     In  3  qr.  15  Ib  ? 

4.  In   12  bales  of  cotton,  averaging  400  Ib.  each, 
how  many  tons,  &c.  ? 

5.  HQW  many  tons  in  9  hogsheads  of  sugar,  con- 
taining an  average  of  1000  pounds  each  ? 

6.  How  many  four-ounce  weights  can  be  made  out 
of  2£  pounds  of  brass  ? 

7.  How  many  seven-pound  packages  of  flour  can 
a  grocer  put  up  from  2  cwt.  ? 

8.  What  cost  2  cwt.  of  cheese,  at  16jc.  a  pound  ? 

9.  Bought  500  Ib.  of  straw,  at  75c.  per  cwt.,  and 
400  Ib.  of  hay,  at  $1.25  per  cwt. ;  what  was  the  bill  ? 

10.  Bought  1  cwt.  of  meat  for  $17.50;  sold  it  at 
22  cents  a  pound.    What  was  the  whole  selling  price  ? 
What  was  the  profit  ? 


80  SEDUCTION. 

SECTION  44.— What  articles  are  weighed  by  Tro y 
Weight  ?  By  Avoirdupois  Weight  ?  By  Apotheca- 
ries' Weight  ?  What  kind  of  pounds  (Troy,  Avoirdu- 
pois, or  Apothecaries')  are  those  in  the  following  Table  ? 

MISCELLANEOUS  TABLE. 

14  pounds,     .     .  1  stone  of  iron  or  lead. 

60  pounds,  .     .     1  bushel  of  wheat. 
100  pounds,     .     .  1  quintal  of  dried  fish. 
100  pounds,  .     .     1  cask  of  raisins. 
196  pounds,     .     .  1  barrel  of  flour. 
200  pounds,  .     .     1  bar.  of  beef,  pork,  or  fish. 

1.  How  many  pounds  in  5J-  stone  ?     In  3^  quin- 
tals of  cod-fish  ?    In  a  quarter  of  a  barrel  of  flour  ? 

2.  Which  is  greater,  7  stone  or  1  cwt.,  and  how 
much  ? 

3.  How  many  bushels  in  3000  Ib.  of  wheat  ? 

4.  What  cost  4  quintals  of  fish,  at  6£c.  a  pound  ? 

5.  What  part  of  1  cwt.  is  1  stone  ? 

6.  What  part  of  a  ton  is  5  cwt.  ?     1200  Ib.  ? 

7.  What  part  of  an  avoirdupois  pound  is  2  ounces  ? 
What   part   of  a  Troy   pound  is  2  ounces?     What 
part  of  an  apothecaries'  pound  is  2  ounces  ? 

8.  What  fraction  of  an  avoirdupois  ounce  is  -fa  of 
a  pound  ?    What  part  of  a  Troy  ounce  ? 

9.  How  do  the  ounce  and  pound  of  Troy  Weight 
compare  with  those  of  Apothecaries'  Weight  ?    Arts. 
They  are  the  same. 

10.  How  do  the  ounce  and  pound  of  Troy  Weight 
compare  with  those  of  Avoirdupois  Weight  ?    Ans. 
The  Troy  ounce  is  greater,  the  Troy  pound  less. 


LONG   MEASUEE.  81 

SECTION  45.— For  what  is  Long  Measure  used  ? 
Ans.  For  measuring  length  or  distance. 


1  inch. 

TABLE  OF  LONG  MEASURE. 


12 

inches  (in.) 

make 

1 

foot, 

. 

. 

ft. 

3 

feet, 

1 

yard 

)  •     • 

.     . 

y<*. 

5| 

:  yards, 

1 

rod, 

.     . 

. 

rd. 

40 

rods, 

1 

furlong,   .     .     . 

fur. 

8 

furlongs, 

1 

mile, 

•     • 

•     • 

mi. 

1.  How  many 

inches  in  t. 

JJft. 

?     In 

2  yd. 

9 

2.  How  many  inches  in  4  yd.  1  ft.  5  in.  ? 

3.  How  many  yards,  &c.,  in  91  ft.  ?   In  80  inches? 

4.  How  many  rods  in  a  mile  ?    How  many  yards  ? 

5.  How  many  feet  in  a  mile  ?     In  half  a  mile  ? 

6.  How  many  miles  in  480  rods  ?    In  18  furlongs  ? 

7.  How  many  feet  in  2  rods  ?     In  99  inches  ? 

8.  How  long  will  it  take  a  person,  walking  at  the 
rate  of  20  rods  a  minute,  to  go  2  miles  ? 

9.  4  inches  make  a  hand.     What  is  the  height  in 
feet  of  a  horse  15£  hands  high? 

10.  What  is  used  in  measuring  drygoods  ?     Ans. 
The  yard  of  long  measure,  divided  into  halves,  quar- 
ters, eighths,  and  sixteenths. 

11.  Three  dress-patterns,  9£,   10^,  and    8-J   yards 
long,  were  cut  from  a  piece  containing  36  yards  ;  how 
many  yards  were  left  ?     What  did  the  three  dresses 
cost,  at  60  cents  a  yard  ? 

12.  How  many  half-yards  of  velvet  can  be  cut 
from  a  piece  9|-  yards  long  ?     How  many  eighths  ? 


1  inch. 


82  REDUCTION. 

SECTION  46.— 1.  For  what  is  Square  Measure 
used  ?  Ans.  For  measuring  surfaces  ;  such  as  land, 
walls,  floors,  &c. 

2.  What  is  a  Square?    Ans.  A       A  SQUARE  INCH. 
Square  is   a  figure   that   has  four 

equal  sides  perpendicular  one  to 
another — that  is,  leaning  no  more 
to  one  side  than  to  the  other. 

3.  What    is     a    Square    Inch? 
Ans.   A    square    whose    sides   are 
each  an  inch  long. 

TABLE  OF  SQUARE  MEASURE. 

144    square  inches  (sq.  in.),  1  square  foot,  sq.  ft. 

9    square  feet,  1  square  yard^  sq.  yd. 

30J  square  yards,  1  square  rod,  sq.  rd. 

40    square  rods,  1  rood,    .     .     R. 

4    roods,  1  acre,  ...  A. 

640    acres,  1  square  mile,  sq.  mi. 

4.  How  many  square  rods  in  2  A.  3  R.  ? 

5.  What  part  of  a  square  foot  is  36  sq.  in.  ? 

6.  How  many  square  feet  in  7£  sq.  yd.  ? 

7.  Reduce  £  sq.  rd.  to  square  yards. 

8.  How  many  80-acre  farms  will  1  sq.  mi.  make  ? 

9.  Reduce  8  square  rods  to  square  yards. 

10.  What  part  of  a  rood  is  ^  of  an  acre  ? 

11.  How  many  square  rods  in  J-  A.  and  f  R.  ? 

12.  A  person,  having  20  A.  of  land,  sold  it  off  in 
lots  of  10  sq.  rd.     How  many  lots  did  it  make? 

13.  What  will  it  cost  to  plaster  288  sq.  ft,,  at  25 
cents  a  square  yard  ? 


CUBIC   MEASURE.  83 

SECTION  47.— 1.  For  what  is  Cubic  Measure 
used  ?  Ans.  For  measuring  bodies,  which  have 
length,  breadth,  and  thickness ;  such  as  timber,  earth, 
boxes,  &G. 

2.  What  is  a   Cube?     Ans.  A  Cube  is  a  body 
bounded  by  six  equal  squares. 

3.  What  is  a  Cubic  Inch  ?    Ans.  A  cube,  one  inch 
long,  one  inch  broad,  and  one  inch  thick.     Each  of  its 
six  sides  is  a  square  inch. 

4.  W  h  a  t    is    a 
Cord  ?    Ans.  A  Cord 
is  a  pile  of  wood  8  ft. 
long,  4  ft.  wide,  and  4  ^ 
ft.  high. 

One  foot  in  length  of  such  a  pile  is  called  a  Cord 
Foot. 

TABLE  OF  CUBIC  MEASURE. 

1728  cubic  inches  (cti.  in.),      1  cubic  foot,  cu.  ft. 
27  cubic  feet,  1  cubic  yard,  cu.  yd. 

40  cu.  ft.  of  round,  or        )  , 

n      ,, ,  .    -,       >  1  ton  or  load,  T. 

50  cu.  ft.  ol  hewn  timber, ) 

16  cubic  feet,  1  cord  foot,     cd.  ft. 

8  cord  feet,  1  cord,     .     .  Cd. 

5.  How  many  cords  in  a  pile  of  wood,  24  feet  long, 
4  feet  wide,  and  4  feet  high  ? 

MODEL.  1  cord  being  8  ft.  long,  4  ft.  wide,  and  4  ft.  high,  there 
are  as  many  cords  as  8  x  4  x  4  is  contained  times  in  24  x  4  x  4,  or  3. 
Ans.  3  Cd. 

6.  How  many  cords  in  a  pile  of  wood,  36  feet  long, 
4  feet  wide,  and  8  feet  high  ? 


84  REDUCTION. 

SECTION  48,— For  what  is  Liquid  or  Wine  Measure 
used  ?     Ans.  For  measuring  liquids  generally. 

TABLE  OF  LIQUID  MEASURE. 

4  gills  (gi.)  make  1  pint,    .     .     .     .     pt. 

2  pints,  1  quart,   .     .     .     .  qt. 

4  quarts,  1  gallon,     .     .     .     gal. 

3 1|-  gallons,  1  barrel, .     .     .     .  bar. 

2  barrels  (63  gal.),  1  hogshead,    .     .     hhd. 

2  hogsheads,  1  pipe,     .     .     .     .pi. 

2  pipes,  1  tun,    ....     tun. 

1.  How  many  gallons  in  45  pt.  ?     In  150  gi.  ? 

2.  How  many  quarts  in  101  gi.  ?     In  2  bar.  ? 

3.  Reduce  5  gal.  1  pt.  1  gi.  to  gills. 

4.  Reduce  f  of  a  gallon  to  pints.     To  gills. 

5.  Reduce  3  qt.  1  pt.  to  gills. 

6.  What  part  of  a  gallon  is  1  qt.  ?     1  pt.  ? 

7.  Add  £  gal.,  %  qt.,  and  |  pt. 

8.  How  many  gallons  in  5  hhd.  ?    In  8  bar.  ? 

9.  If  a  tumbler  holds  half  a  pint,  how  many  times 
will  1  gallon  of  water  fill  it  ? 

10.  Bought  5  gal.  of  oil  for  $10,  what  is  the  price 
per  quart  ? 

11.  Bought  a  pint  of  milk  for  4  cents;  at  what 
rate  is  that  per  gallon  ? 

12.  What  cost  18  quarts  of  wine,  at  16s.  per  gal.? 

13.  3  pt.  of  molasses  having  been  sold  out  of  2 
gal.,  what  is  the  remainder  worth  at  80c.  a  gal.  ? 

14.  If  5  bottles  hold  a  gallon  of  wine,  what  part 
of  a  quart  does  each  hold  ? 


DRY    MEASURE.  85 

SECTION  49.— For  what  is  Dry  Measure  used  ? 
Ans.  For  measuring  grain,  vegetables,  salt,  coal,  and 
other  articles  not  liquid. 

TABLE  OF  DRY  MEASURE. 

2  pints  (pt.)  make  1  quart,  .     .     .     .     qt. 

8  quarts,  1  peck,     ....  pk. 

4  pecks,  1  bushel,     .     .     .     bu. 

36  bushels,  1  chaldron,   .     .     .  chal. 

1.  How  many  bushels  in  165  qt.  ?     In  191  pt.  ? 

2.  Reduce  1  bu.  5  qt.  to  quarts. 

3.  How  many  bushels  in  5£  chaldrons  ? 

4.  How  many  pints  in  3  bu.  3  pk.  3  qt.  ? 

5.  Reduce  -f^  of  a  peck  to  lower  denominations. 

MODEL.  1  pk.  =  8  qt.  Hence,  -f-0-  of  a  peck  is  -,90-  of  8  qt.,  or 
^  of  1  qt,  which  equals  7-J-  qt.  1  qt.=2  pt.  Hence,  J  of  a  quart 
is  £  of  2  pt.,  or  |  of  1  pt.  Ans.  7  qt.  f  pt. 

6.  Reduce  -^  of  a  bushel  to  lower  denominations. 

7.  Reduce  f  of  a  peck  to  lower  denominations. 

8.  How  many  bushels  in  5f  chaldrons  ? 

9.  If  a  chaldron  of  coal  costs  $12,  what  is  the  price 
per  bushel  ? 

10.  How  many  baskets,  holding  2^  pecks  each,  will 
5  bushels  of  peaches  fill  ? 

11.  What  part  of  a  bushel  is  half  a  peck? 

12.  What  part  of  a  quart  is  fa  of  a  peck  ? 

13.  6  quarts  are  -|  of  how  many  pecks  ? 

14.  If  a  horse  is  fed  6  qt.  of  oats  a  day,  how  long 
will  it  take  him  to  consume  3  bushels  ? 

15.  If  a  family  consume  4  bushels  of  coal  a  day, 
how  long  will  it  take  them  to  use  3^  chaldrons  ? 


86  REDUCTION. 

SECTION  50.— 1.  What  are  the  natural  divisions 
of  time  ?  Am.  The  Year,  in  which  the  Earth  revolves 
round  the  Sun  ;  and  the  Day,  in  which  it  turns  on  its 
axis. 

2.  How  is  the  year  divided  ?     Ans.  Into  twelve 
calendar  months,  differing  in  length. 

3.  How  is  the  day  divided  ?     Ans.    Into  hours, 
minutes,  and  seconds. 

TABLE  OF  TIME  MEASURE. 

60  seconds  (sec.)  make  1  minute,    .     .     min. 

60  minutes,  1  hour,     .     .     .  h. 

24  hours,  1  day,     .     .     .     da. 

7  days,  1  week,    .     .     .  wk. 

365  days  or  )  _     <     _ 
12  calendar  months, ) 

366  days,  1  leap  year. 

100  years,  1  century,     .     .  cen. 

4.  Learn  the  names  of  the  calendar  months,  and 
the  number  of  days  they  contain  : — 


DATS. 


/st  month,  January,  31. 

2d   month,  February,  28. 

3d   month,  March,  31. 

4th  month,  April,  30. 

5th  month,  May,  31. 

6th  month,  June,  3Q. 


DATS. 


7th  month,  July,  31. 

8th  month,  August,  31. 

9th  month,  September,  30. 

10th  month,  October,  31. 

llth  month,  November,  30. 

12th  month,  December,  31. 


5.  How  many  hours  in  the  month  of  April  ? 

6.  How  many  minutes  in  a  day  ?     In  2  days  3  h.  ? 

7.  How  many  seconds  in  2  h.  ?     In  1  h.  10  min.  ? 

8.  How  many  weeks,  &c.,  in  240  hours? 


PAPER  MEASURE.  8  7 

SECTION  51.— TABLE  OP  PAPER  MEASURE. 

24  sheets  make  1  quire. 
20  quires,  1  ream. 

2  reams,  1  bundle. 

5  bundles,         1  bale. 


TABLE  OF  COLLECTIONS  OF  UNITS. 

12  units  make  1  dozen,  doz. 
12  dozen,          1  gross. 
12  gross,  1  great  gross. 

20  units,  1  score. 

1.  How  many  sheets  in  a  ream  of  paper  ? 

2.  How  many  sheets  in  8f  quires  ?     In  f  of  a  ream  ? 

3.  How  many  quires  will  240  sheets  make? 

4.  How  many  bundles  will  100  quires  make  ? 

5.  What  part  of  a  ream  is  37  sheets  ?     15  quires  ? 

6.  What  part  of  a  gross  is  5  dozen  ?     1  score  ? 

7.  How  many  units  in  a  great  gross  ?    In  11  gross  ? 

8.  If  a  box  of  pens  holds  a  gross,  how  many  pens 
are  there  in  5  boxes  ?     At  ic.  a  pen,  what  will  each 
box  cost  ? 

9.  If  a  person  buys  a  ream  of  paper  for  $3,  and  re- 
tails it  at  1  cent  a  sheet,  how  much  profit  will  he  make  ? 

10.  If  5  dozen  buttons  are  sold  out  of  a  gross  that 
cost  $2.40,  how  much  are  what  remain  worth  ? 

11.  Bought  7  quires  of  foolscap,  12  sheets  of  letter- 
paper,  and  2|-  quires  of  note-paper ;  what  was  it  worth, 
at  the  rate  of  $3.20  a  ream  ? 

12.  How  many  units  in  4  great  gross? 


REDUCTION. 

SECTION  52.— 1.  Which  is  the  shortest  month  ? 

2.  How  many  days  has  February  in   leap-year? 
Ans.  29. 

3.  Which  years  are  leap-years  ?     Ans.  Those  that 
can  be  exactly  divided  by  4,  except  such  of  the  even 
hundreds  as  can  not  be  exactly  divided  by  400. 

4.  Name  the  leap-years  between  1868  and  1901. 

5.  From  January  1st  to  February  1st  how  many 
months  ?     From  Jan.  3d  to  Feb.  3d  ?     From  Jan.  7th 
to  Oct.  7th? 

6.  From  Jan.  5th  to  March  5th  how  many  months  ? 
From  Feb.  1st  to  Aug.  1st  ?   From  June  2d  to  Nov.  2d  ? 

7.  From  April  6th  to  Oct.  6th  how  many  months  9 
From  Aug.  9th  to  Nov.  9th  ? 

8.  What  part  of  a  year  is  2  months  ?     6  months  ? 
8  months  ?     3  months  ?     9  months  ?     5  months  ? 

9.  How  many  months  in  f  of  a  year  ?     In  -£%  of  a 
year  ?     In  T\  of  a  year  ?     In  f  of  a  year  ? 

10.  In  business  calculations,  how  many  days  are 
generally  allowed  to  a  month  ?     Ans.  30  clays. 

11.  In  business  calculations,  what  part  of  a  month 
would  we  call  1  day  ?     2  days  ?     20  days  ?     15  days  ? 
5  days  ?     24  days  ?     1 0  days  ? 

12.  In  business  calculations,  what  part  of  a  year  is 
a  day  considered  ?     Ans.  -^  of  T1g-,  or of  a  year. 

13.  What  part  of  a  year  is  5  days  considered? 

14.  What  part  of  a  year  is  a  day  really?     What 
part  is  five  days  ? 

15.  How  many  calendar  months  in  a  century  ? 

16.  Which  of  the  months  have  31   days?     How 
many  days  has  the  5th  month  ?     The  llth  month  ? 


THE   METKIC   SYSTEM.  89 


CHAPTER    EIGHTH. 

THE       METRIC        SYSTEM. 

SECTION  53.— 1.  What  is  the  Metric  System? 
Am.  A  system  of  weights  and  measures  in  which  it 
takes  10  of  a  lower  denomination  to  make  1  of  the 
next  higher. 

2.  Where  is  the  Metric  System  used?    Am.  In 
France,  Belgium,  and  other  countries  of  Europe.     Its 
use  is  also  authorized  by  law  in  the  United  States. 

3.  What  is  the  unit  of  length  ?     Am.  The  Metre, 
from  which  the  Metric  System  has  its  name. 

4.  How  long  is  a  Metre  ?  Am.  About  39^^  inches. 

5.  How  are  lower  denominations  formed?     Am. 
From  the  metre  other  denominations,   -^  y^-,  and 
ToW  as  greatj  are  formed  with  the  prefixes  deci  (pro- 
nounced des'e),  centi,  and  milli. 

6.  How  are  higher  denominations  formed  ?     Ans. 
Denominations  10,  100,  1000,  and  ten  thousand  times 
as  great  as  the  metre,  are  formed  with  the  prefixes 
deca,  Recto,  kilo,  and  myria. 

MEASURES  OF  LENGTH. 


iu  mu  iimetres  r 

naK( 

)  l  cen  timetre 

—  TOT  incn- 

10  centimetres 

a 

1  dec'imetre 

=  319^s-  inches. 

10  decimetres 

u 

1  ME'TKE 

=  39TVu-  inches. 

10  metres 

u 

1  dec'ametre 

=  32  ft,  9TV  in. 

10  decametres 

a 

1  hec'tometre 

=  328  ft.  1  in. 

10  hectometres 

u 

1  kil'ometre 

=  3280  ft.  10  in. 

10  kilometres 

a 

1  myr'iametre 

=  6T2o^  miles. 

*  The  equivalents  given  in  these  Tables  are  not  exact,  hut  nearly  so. 


00  THE    METRIC    SYSTEM. 

7.  How  many  centimetres  in  a  metre?     In  half  a 
metre  ?     In  5  metres  ?   In  17  metres  ?   In  a  decimetre  ? 

8.  How  many  kilometres  in  4000  metres  ? 

9.  How  many  kilometres  in  500  decametres? 

10.  What  part  of  a  decametre  is  a  decimetre  ?     A 
metre  ?     5  metres  ?     8  metres  ? 

11.  What  part  of  a  metre  is  a  decimetre  ?  A  centi- 
metre ?    What  part  of  a  hectometre  is  a  decametre  ? 

12.  How  many  metres  in  15  decimetres  ?    In  £  of  a 
decametre?  In  250  centimetres  ?  In  J  of  a  hectometre  ? 

13.  Is  a  metre  more  or  less  than  a  yard  ?     How 
many  inches  more  ? 

14.  About  how  many  metres  equal  1  rod  ? 

15.  What  cost  4  metres  of  cloth,  at  $3£  a  metre  ? 

16.  If  a  person  sells  7  decimetres  of  velvet  from  a 
piece  containing  3  metres,  how  much  is  what  remains 
worth,  at  $10  a  metre  ? 

17.  How  many  kilometres  of  fence  will  be  required, 
to  surround  a  square  field  500  metres  on  each  side  ? 

18.  At  75c.  a  metre,  what  will  it  cost  to  fence  a 
square  field,  30  metres  on  each  side  ? 


SECTION  54,— 1.  What  is  the  unit  of  surface  ? 
Ans.  The  Are  (pronounced  a/r),  a  square  whose  side 
is  10  metres,  and  which  equals  119|  square  yards. 

MEASURES  OF  SURFACE. 

The  cen'tiare  is  1  square  metre,  or  1550  square  inches. 
100  centiares  make  1  ARE         =  119f  sq.  yd. 
100  ares  "     1  hec'tare  =  2-^  acres. 


THE   METRIC   SYSTEM.  91 

2.  What  is  the  unit  of  capacity  ?    Ans.  The  Litre 
(pronounced  le'tur),  a  cube  equal  to  about  T9¥  of  a 
quart  of  dry  measure. 

3.  How  are  other  denominations  formed?     Ans. 
As  before,  by  means  of  the  prefixes  deci,  centi,  and 
milli  for  lower  denominations,  and  deca,  hecto,  and 
kilo  for  higher  ones. 

MEASUEES  OF  CAPACITY. 

Dry  Measure.        Liquid  Measure. 

10  nnTlilitres,     1  cen'tilitre     =  f    cu.  in.   =         J-    fluid  oz. 
10  centilitres,     1  dec'ilitre      =6TV  cu.  in.  =        f    gill. 
10  decilitres,      1  LI'TRE  =  T97  quart     =       1^  quart. 

10  litres,  1  dec'alitre     =ly    peck      =         2f  gallons. 

10  decalitres,      1  hec'tolitre    =2f    bushels  =     26f£  gallons. 
10  hectolitres,    1  kil'olitre      =1 A  cu.  yd.  =264^  gallons. 
The  kilolitre  (1  cubic  metre),  when  used  in  measuring 
wood,  is  called  the  STERE,  and  equals  about  J^  of  a  cord. 
10  steres  make  1  dec'astere  =  about  2|  cords. 

4.  How  many  ares  in  550  centiares  ?  In  5  hectares  ? 

5.  Are  2  hectares  more  or  less  than  5  acres,  and 
how  much  ? 

6.  What  part  of  an  are  is  65  centiares  ? 

7.  What  part  of  a  litre  is  5  centilitres  ?     7  decili- 
tres ?     2  millilitres  ? 

8.  How  many  litres  in  1 7 J  decalitres  ?    In  fa  of  a 
hectolitre  ?    In  70  decilitres  ? 

9.  What  is  the  cost  of  1^  hectolitres  of  molasses, 
at  12  cents  a  litre? 

10.  How  many  litres  of  molasses,  at  11  cents  a  litre, 
should  be  given  in  exchange  for  3  decalitres  and  3 
litres  of  vinegar,  at  8  cents  a  litre  ? 


92  THE    METRIC    SYSTEM. 

11.  If  a  person  bought  2  hectolitres  of  molasses, 
and  sold  10  decalitres,  what  was  the  rest  worth,  at 
12J  cents  a  litre  ? 

12.  If  one  stere  equals  J-J-  of  a  cord,  how  many 
steres  make  a  cord  ? 

MODEL.  If  1  stere  equals  -Ji  of  a  cord,  it  will  take  as  many 
steres  to  make  a  whole  cord,  or  JJ,  as  -J-^  is  contained  times  in  f  £ ; 
that  is  (rejecting  the  denominators),  as  11  is  contained  times  in  40, 
or  3-^-.  Ans.  3  fr  steres. 

13.  If  1  litre  equals  T9F  of  a  quart,  dry  measure, 
how  many  litres  make  a  quart  ? 

14.  Bought  a  decastere  of  wood,  and  sold  4  steres 
of  it  for  $8  ;  how  much  was  that  a  stere  ?     How  much 
was  what  remained  worth  at  the  same  rate  ? 

15.  At  $1.75  a  stere,  what  is  the  cost  of  2  decasteres 
of  wood  ? 

16.  A  man  bought  3  hectolitres  of  potatoes  for  89. 
He  sold  them  for  40  cents  a  decalitre.     Did  he  gain  or 
lose,  and  how  much  ? 

17.  Reduce  1  litre  1  decilitre  1  centilitre  to  centi- 
litres. 

18.  The  kilolitre  being  a  cubic  metre,  how  long  in 
inches  is  each  side  of  the  cube  it  represents  ? 

19.  If  1  decastere  equals  2|  cords,  how  many  cords 
do  5  decasteres  equal  ?     5^  decasteres  ? 

20.  How  many  decasteres  do  19£  cords  equal? 

21.  If  a  litre  equals  T\  of  a  quart,  dry  measure,  how 
many  litres  are  there  in  45  quarts  ? 

22.  What  prefixes  are  used  with  the  metre,  <fcc.,  to 
form  lower  denominations  ?     What  prefixes  are  used, 
to  form  higher  denominations  ? 


THE   METRIC    SYSTEM.  93 

SECTION  55. — 1.    What  is  the  unit   of  weight? 
Ans.  The  Gram,  which  equals  about  15T4¥^  grains. 

WEIGHTS. 


10  milligrams, 

1  cen'tigram  = 

,V5o  grain. 

10  centigrams, 

1  dec'igram    = 

IfJ  grains. 

10  decigrams, 

1  GRAM                = 

15T4oV  grains. 

10  grams, 

1  dec'agram    = 

•fo  oz.  av. 

10  decagrams, 

1  hec'togram  = 

3J  oz.  av. 

10  hectograms, 

1  kil'ogram     = 

2£  Ib.  av. 

10  kilograms, 

1  myr'iagram  = 

22  Ib.  av. 

10  myriagrams, 

1  quintal         = 

220f£  Ib.  av. 

10  quintals, 

1  tonneau       = 

1TV  tons. 

2.  Which  is  greater,  the  common  quintal  or  the 
quintal  of  the  Metric  System  ?     How  many  pounds 
greater  ? 

3.  Which  is  greater,  a  ton  or  a  tonneau  ? 

4.  If  a  decagram  equals  -^  of  an  ounce  avoirdu- 
pois, how  many  decagrams  will  make  an  ounce? 

5.  How  many  kilograms  in  a  quintal  ?     In  a  ton- 
neau ?    In  1  tonneau  9  quintals  ?     In  4500  grams? 

6.  How  many  grams  in   7£  hectograms  ?    In  900 
centigrams  ?     In  900  decagrams  ?     In  900  decigrams? 

7.  If  1  kilogram  equals  2|  Ib.,  when  butter  sells  at 
$1.10  a  kilogram,  how  many  cents  a  pound  is  it  ? 

8.  What  price  per  kilogram  is  equivalent  to  40 
cents  a  pound  ? 

9.  How  many  powders  of  one  gram  each  can  be  put 
up  from  ^  of  a  hectogram  ? 

10.  What  part  of  a  tonneau  is  a  kilogram  ?     4 
myriagrams  ? 


THF 

UNIVERSITY 


04  COMPOUND   ADDITION. 


CHAPTER     NINTH. 

THE     COMPOUND     RULES. 

SECTION  56.— 1.  What  is  a  Compound  Number? 
Am.  A  number  containing  different  denominations; 
as,  1  foot  2  inches. 

2.  What  is  the  addition   of  compound   numbers 
called  ?    Ans.  Compound  Addition. 

3.  What  is  the  sum  of  £2  17s.  6d.  and  £3  8s.  4d.  ? 
MODEL.     The  sum  of  the  pence  is  4d.  +  6d.,  or  lOd.     The  sum  of 

the  shillings  is  8s.  +  17s.,  or  25s.,  which  equals  £1  5s.  The  sum  of 
the  pounds  is  £3 +  £2,  which,  with  the  £1  of  the  last  sum,  makes 
£6.  Ans.  £6  5s.  lOd. 

4.  What  is  the  sum  of  15s.  9d.  and  3s.  6d  ? 

5.  What  is  the  sum  of  £l  3s.  lOd.  and  £5  9s.  4d.  ? 

6.  A  boy  throws  3  pk.  6  qt.  of  potatoes  into  a  bar- 
rel already  containing  2  pk.  3  qt.     How  many  bushels, 
&c.,  are  then  in  the  barrel  ? 

7.  How  much  jalap  in  two  bottles,  the  one  con- 
taining 4  oz.  3  dr.  2  sc.,  and  the  other  8  oz.  2  dr.  1  sc.  ? 

8.  To  3  gal.  1  qt.  1  pt.  of  alcohol  is  added  1  gal. 
3  qt.  1  pt.  of  water ;  how  much  is  there  of  the  mixture  ? 

9.  In  the  morning  a  person  walks  1  mi.  1  fur.  20  rd., 
in  25  minutes;  in  the  afternoon  he  walks  4  mi.  3  fur. 
30  rd.,  in  1  h.  35  min.  40  sec.     What  distance  does  he 
walk  altogether,  and  in  what  time  ? 

10.  A  grocer  mixes  three  kinds  of  tea  ;  5  Ib.  11  oz. 
of  the  first  kind,  6  Ib.  7  oz.  of  the  second,  and  3  Ib.  10 
oz.  of  the  third.     How  many  pounds  in  the  mixture  ? 


COMPOUND   ADDITION.  95 

11.  A  person,  having  53  A.  1  R.  25  sq.  rd.  of  land, 
bought  7  A.  20  sq.  rd.  more ;  how  much  had  he  then  ? 

12.  Bought  some  lace  for  $3.17,  and  some  calico  for 
$2.85  ;  what  was  the  amount  of  the  bill? 

MODEL.  The  amount  of  the  bill  was  the  sum  of  $3.17  and  $2.85, 
or  |6.02.  Am.  $6.02. 

NOTE.  In  the  case  of  Federal  Money,  add  as  in  simple  addition; 
if  the  sum  is  in  cents,  cut  off  the  two  right-hand  figures  for  cents, 
and  what  remains  on  the  left  will  be  dollars. 

13.  If  I  spend  $1.10  for  paper,  $2.80  for  books,  and 
25  cents  for  pens,  how  much  do  I  spend  in  all? 

14.  Bought  butter  for  $1.14  ;  cheese  for  2 7c. ;  lard 
for  32c. ;  sugar  for  $1  ;  what  did  the  bill  amount  to  ? 

15.  A  person  received  three  telegrams.     The  first 
cost  90c.,  the  second  $1.45,  and  the  third  85c.     What 
was  the  cost  of  all  three  ? 

16.  A  carpenter  laid  out  $1.25  for  nails,  $2.90  for 
bolts,  and  75c.  for  screws  ;  how  much  was  his  bill  ? 

NOTE.  Sometimes  the  items  can  be  combined  in  a  particular 
order  with  advantage.  Thus,  in  the  last  example,  the  sum  of  $1.25 
and  75c.  is  $2 ;  $2  and  $2.90  make  $4.90. 

17.  A  storekeeper  took  in  40c.  in  the   morning, 
$5.33  in  the  afternoon,  and  $4.60  in  the  evening ;  how 
much  did  he  take  in  during  the  day  ? 

18.  John  Ray  bought  of  T.  Kipp  a  dozen  slates  for 
$2.16,  some  envelopes  for  $1.72,  pencils  for  84c.,  and 
diaries  for  $4.28.     How  much  was  Ray's  bill? 

19.  A  collection  having  been  taken  up,  the  plate  was 
found  to  contain  18  cents,  21  two-cent  pieces,  10  three- 
cent  pieces,  10  five-cent  stamps,  6  10-cent  stamps,  and 
4  25-cent  stamps  ;  how  much  was  collected  ? 


96  COMPOUND   SUBTRACTION. 

SECTION  57.— 1.  What  is  the  subtraction  of  com- 
pound  numbers  called  ?  Ans.  Compound  Subtraction, 

2.  A  jeweller,  who  had  1  Ib.  11  oz.  4  pwt.  3  gr.  of 
gold,  used  10  oz.  15  pwt.  20  gr. ;  how  much  had  he  left  ? 

MODEL.  He  had  left  the  difference  between  1  Ib.  11  oz.  4  pwt.  3 
gr.  and  10  oz.  15  pwt.  20  gr.  20  gr.  can  not  be  taken  from  3  gr. ; 
we  therefore  take  1  of  the  next  higher  denomination  (1  pwt.),  re- 
duce it  to  grains  (24),  add  it  to  3  gr.,  and  then  subtract.  24  gr.  +  3 
gr.=27  gr.  27  gr.-20  gr.=7  gr. 

To  balance  the  pennyweight  thus  added,  we  now  add  1  pwt.  to 
the  15  pwt.  to  be  subtracted,  making  16  pwt.  16  pwt.  can  not  be 
taken  from  4  pwt. ;  we  therefore  take  1  of  the  next  higher  denom- 
ination (1  oz.),  reduce  it  to  pennyweights  (20),  add  it  to  the  4  pwt., 
and  then  subtract.  20  pwt. +4  pwt. =24  pwt.  24  pwt— 16  pwt.  — 
8  pwt. 

To  balance  the  ounce  thus  added,  we  now  add  1  oz.  to  the  10  oz. 
to  be  subtracted,  making  11  oz.  11  oz.— 11  oz.=0  oz.  1  Ib.— 0= 
1  Ib.  Ans.  1  Ib.  8  pwt.  7  gr. 

3.  From  a  bin  containing  10  bu.  3  pk.  1  qt.  of  oats, 
a  person  took  3  bu.  2  pk.  6  qt.     How  much  was  left  ? 

4.  If  from  1  day  we  take  19  hours  20  min.,  how 
much  time  will  remain  ? 

5.  A  person  having  a  bill  of  £5  10s.  to  pay,  has 
only  £3  7s.  9d.     How  much  does  he  lack  ? 

6.  How  much  of  my  fence  remains  to  be  built,  if 
there  is  15  rd.  5  yd.  to  be  built  in  all,  and  11  rd.  2  yd. 
1  ft.  is  completed  ? 

7.  To  10  gal.  1  qt.  of  whiskey  a  liquor-merchant 
adds  3  qt.  of  water,  and  then  sells  7  gal.  2  qt.  1  pt.  of 
the  mixture.     How  much  remains  ? 

8.  A  hardware-merchant,  having  on  hand  1  cwt. 
10  Ib.  of  lead,  buys  12  cwt.  20  Ib.  more,  and  then  sells 
10  cwt.  15  Ib.     How  much  has  he  left  ? 


COMPOUND    SUBTRACTION.  97 

9.  How  many  years,  months,  and  days,  from  No- 
vember 9th,  1867,  to  May  1st,  1868  ? 

NOTE.  November  is  the  llth  month,  May  the  5th.  We  there- 
fore  take  1867  years  11  months  9  days,  which  is  the  earlier  date, 
from  1868  years  5  months  1  day.  Allow  30  days  to  the  month. 
Ans.  5  months  22  days. 

10.  How  many  months  and  days  from   January 
15th  to  October  3d  of  the  same  year? 

11.  How  many  months  and  days  from  July  20th  to 
December  29th  of  the  same  year  ? 

12.  How  many  months  and  days  from  July  20th  to 
December  29th  of  the  preceding  year  ? 

13.  How  many  years,  months,  and  days,  from  Au- 
gust 25th,  1865,  to  March  13th,  1868  ? 

14.  Some  mats  were  bought  for  $18.60,  and  sold  at 
a  loss  of  $3.85  ;  how  much  did  they  bring  ? 

MODEL.  If  they  were  bought  for  $18.60  and  sold  at  a  loss  of 
$3.85,  they  brought  $18.60— $3.85,  or  $14.75.  Ati*.  $14.75. 

NOTE.  As  in  the  addition  of  Federal  Money,  so  in  subtraction, 
multiplication,  and  division ;  if  the  result  is  in  cents,  cut  off  the  two 
right-hand  figures  for  cents,  and  what  is  left  will  be  dollars. 

15.  A  owes  -B  $25.20,  and  gives  him  on  account 
$17.75.     If  he  pays  the  balance  with  a  ten-dollar  bill, 
how  much  change  should  he  receive  ? 

16.  What  is  the  profit  on  goods  bought  for  $107.50, 
and  sold  for  $119.25  ? 

17.  Some  goods,  bought  for  $74.50,  are  sold   for 
$69.66.     Does  the  owner  gain  or  lose,  and  how  much  ? 

18.  On  a  bill  of  $48,  C  has  paid  $29.25.     He  now 
pays  the  balance  with  a  twenty-dollar  bill ;  how  much 
change  should  he  receive  ? 


98  COMPOUND   MULTIPLICATION. 

SECTION  58.— 1.  What  is  the  multiplication  of  a 
compound  number  called  ?  Ans.  Compound  Multipli- 
cation. 

2.  How  many  pounds,  &c.,  in  four  packages,  each 
containing  3  Ib.  5  oz.  4  dr.  ? 

MODEL.  Four  packages,  each  containing  3  Ib.  5  oz.  4  dr.,  will 
contain  4  times  3  Ib.  5  oz.  4  dr.  4  times  4  dr.  is  16  dr.,  or  1  oz., 
4  times  5  oz.  is  20  oz.,  and  1  oz.  (the  last  product)  makes  21  oz.,  or 
1  Ib.  6  oz.  4  times  3  Ib.  is  12  Ib.,  and  1  Ib.  (from  the  last  product) 
makes  13  Ib.  Ans.  13  Ib.  5  oz, 

3.  What  cost  a  dozen  Histories,  at  3s.  8d.  each  ? 

4.  If  a  person  gives  his  five  children  each  £1  10s. 
6cl.,  how  much  does  he  give  them  in  all  ? 

5.  What  are  the  contents  of  eight  pitchers,  if  each 
holds  1  qt.  1  pt.  1  gi.  ? 

6.  What  is  the  breadth  of  three  strawberry  beds, 
each  1  yd.  1  ft.  10  in.  broad  ? 

7.  How  much  land  in  7  fields,  each  containing  1  A. 
15  sq.  rd.  ? 

8.  What  cost  8  knives,  at  $1.38  each  ?  Ans.  $11.04. 

9.  What  cost  9  pigs,  at  $5.30  apiece  ? 

10.  What  cost  12  caps,  at  $1.33^  apiece? 

11.  Bought  10  albums,  at  $2.64  each,  and  5  pocket- 
books,  at  75c.  each  ;  what  did  the  bill  come,  to  ? 

12.  Bought  3  pair  of  gloves,  at  $1.05  each ;  4  col- 
lars, at  $3.60  each  ;  and  11  yards  of  calico,  at  25c.  per 
yd.     What  was  the  amount  of  the  bill  ? 

13.  Bought  20  chickens,  at  60c.  each  ;  6  ducks,  at 
75c.  each  ;  and  8  turkeys,  at  $1.60  each.     Paid  on  ac- 
count $16.80  ;  how  much  remained  due?     How  many 
fifty-cent  stamps  will  pay  the  balance  ? 


COMPOUND   DIVISION.  99 

SECTION  59.— 1.  What  is  the  division  of  a  com- 
pound number  called  ?  Ans.  Compound  Division. 

2.  If  a  certain  pipe  fills  a  cistern  in  2  h.  21  min. 
25  sec.,  how  long  will  it  take  4  such  pipes  to  fill  it  ? 

MODEL.  If  a  certain  pipe  fills  a  cistern  in  2  h.  21  rain.  25  sec., 
four  such  pipes  will  fill  it  in  £  of  2  h.  21  min.  25  sec. 

4  is  not  contained  in  2  h. ;  we  therefore  reduce  2  h.  to  minutes 
(120  min.),  which  we  add  to  the  21  min.,  getting  141  min.  £  of  141 
min.r=35  min.,  and  1  min.  remainder.  We  reduce  this  remainder,  1 
min.,  to  seconds  (60  sec.),  and  add  the  result  to  the  25  sec.,  getting 
85  sec.  i  of  85  sec.  =  21£  sec.  Ans.  35  min.  21£  sec. 

3.  If  1  man  can  do  a  piece  of  work  in  4  h.  45  min., 
how  long  will  it  take  6  such  men  to  do  it  ? 

4.  Dividing  20  bu.  7  pk.  3  qt.  of  potatoes  into  10 
equal  heaps,  how  much  have  we  in  each  heap  ? 

5.  It  took  James  f  as  long  to  walk  a  certain  dis- 
tance as  it  did  Andrew.     If  it  took  Andrew  1  h.  42 
inin.,  how  long  did  it  take  James  ? 

6.  A  certain  keg  contained  5  gal.  2  qt.  1  pt.  of 
wine  mixed  with  water.     If  ^  of  the  mixture  was 
water,  how  much  wine  was  there  in  the  keg  ? 

7.  A  grocer  had  2  cheeses,  one  of  which  weighed 
|  as  much  as  the  other.     If  the  larger  weighed  14  Ib. 
4  oz.,  what  did  the  smaller  weigh  ?     If  the  smaller 
weighed  14  Ib.  4  oz.,  how  much  did  the  larger  weigh  ? 

8.  If  a  person's  weekly  expenses  average  $40.25, 
how  much  is  that  per  day  ? 

9.  How  much   will  1   portfolio  cost,  if  they  are 
$21.60  a  dozen  ? 

10.  A  person  bought  some  goods  for  $25.80,  and 
sold  them  for  f  of  that  sum  ;  what  was  his  loss  ? 


100  MISCELLANEOUS    EXAMPLES. 

11.  How  many  bags,  holding  1  bu.  1  pk.  2  qt,  each, 
will  11  bu.  3  pk.  2  qt.  of  oats  till  ? 

MODEL.  As  many  bags  as  1  bu.  1  pk.  2  qt.  is  contained  times  in 
11  bu.  3  pk.  2  qt.  1  bu.  1  pk.=5  pk.  5  pk.  2  qt.=42  qt.  11  bu. 
3  pk.=47  pk.  47  pk.  2  qt.=378  qt.  42  qt.  are  contained  in  378 
qt.  9  times.  Ans.  9  bags. 

12.  How  many  books,  costing  2s.  6d.  each,  can  be 
bought  for  £1  7s.  6d.  ? 

13.  How  many  pitchers,  holding  1  qt.   1  pt.   3  gi. 
each,  will  4  gall.  2  qt.   1  pt.  2  gi.  fill  ?    How  many 
will  1  gal.  3  qt.  1  pt.  fill? 

14.  How  many  dishes,  at  $1.10,  can  be  bought  for 
$5.50?     For  $11? 


CHAPTER     TENTH. 

MISCELLANEOUS      EXAMPLES. 

SECTION  60.— 1.  Fifteen  barrels  of  pork,  bought 
for  $23  a  barrel,  were  sold  for  $23.62.  What  was  the 
profit  on  the  lot  ? 

2.  Five  yards  of  cloth  were   bought   for  $32.50  ; 
being  damaged,  they  were  sold  at  a  loss  of  $2.75. 
What  did  they  bring  per  yard  ? 

3.  At  what  price  each  must  I  sell  12  tables,  in  order 
to  gain  $15,  if  the  whole  were  bought  for  $60  ? 

4.  A  merchant  bought  25  yd.  of  silk  for  $52,50. 
He  sold  15  yd.  of  it  at  §2£  a  yard,  and  the  rest  for 
$2.60  a  yard.     Did  he  gain  or  lose  on  the  whole,  and 
how  much  ? 


MISCELLANEOUS    EXAMPLES.  101 

5.  Bought  10  stoves  for  $150;  at  what  price  must 
they  be  sold  apiece,  in  order  to  make  the  cost  of  one 
stove  on  the  whole  ? 

6.  If  a  person  buys  a  ton  of  hay  for  $20,  and  sells 
5  cwt.  at  80c.  a  cwt.,  and  7  cwt.  at  90c.,  what  must  he 
sell  the  rest  for  per  cwt.,  so  as  not  to  lose  on  the  whole  ? 

7.  A  woman  buys  some  oranges  at  2  cents  apiece. 
She  sells  half  of  them  at  the  rate  of  2  for  3  cents,  and 
the  rest  at  the  rate  of  3  cents  apiece.     She  makes  a 
dollar  ;  how  many  oranges  had  she  ? 

8.  Three  persons  bought  some  goods  for  $125.90, 
and  sold  them  for  $150.41.     What  was  the  profit  of 
each? 

9.  Eight  desks  were  bought  for  $25  each,  and  sold 
for  $23.75  each.    How  much  did  the  owner  lose? 

10.  A  person,  having  bought  3  barrels  of  flour  for 
$28.80,  let  his  brother  have  f  of  a  barrel  at  cost;  how 
much  was  that  ? 

11.  A  farm  was  sold  for  $1800,  which  was  -f  of  its 
cost.     What  was  the  loss  ? 

12.  How  many  bushels,  &c.,  of  apples,  worth  80 
cents  a  bushel,  should  be  given  for  3  yards  of  cloth, 
worth  $3.50  a  yard  ? 

13.  £1  is  worth  $4.86^;  at  this  rate,  how  much 
is  one  shilling  worth  ?     One  penny  ? 

14.  A  franc  is  worth  19T3^c.,  a  shilling  22c. ;  how 
many  francs  are  equal  in  value  to  1  shilling  ? 

15.  Three  men  bought  a  horse  for  $125.     After 
keeping  him  2  months  15  days,  during  which  time 
they  paid  $20  a  month  for  stabling,  they  sold  him  for 
$130.     What  was  each  man's  share  of  the  loss? 


102  MISCELLANEOUS   EXAMPLES. 

16.  If  7  oranges  are  equal  in  value  to  1  pine-apple, 
and  3  pine-apples  to  28  lemons,  how  many  oranges  are 
8  lemons  worth  ? 

MODEL.  If  28  lemons  are  worth  3  pine-apples,  1  lemon  is  worth 
•£s  of  3  pine-apples,  or  5\  of  1  pine-apple ;  and  8  lemons  are  worth 
8  times  $,-,  or  f -J,  or  f,  of  a  pine-apple.  But  7  oranges  equal  1 
pine-apple  hi  value ;  hence,  if  8  lemons  are  worth  f  of  a  pine-apple, 
they  must  be  worth  f  of  1  oranges,  or  6  oranges.  Ans.  6  oranges. 

17.  If  2  apples  are  worth  as  much  as  1  pear,  and  3 
melons  as  much  as  10  pears,  how  many  melons  should 
be  given  for  80  apples  ? 

18  A  boy,  having  sold  1  bu.  2  pk.  of  blackberries 
at  lOc.  a  quart,  took  in  part  payment  1  pk.  2  qt.  of 
timothy  seed,  at  $2.40  a  bushel.  How  much  was  still 
due  him  ? 

19.  How  many  guineas  are  £10  10s.  equal  to  ? 


SECTION  61.— 1.  How  long  will  it  take  15  men  to 
do  a  piece  of  work,  if  5  men  can  do  it  in  9  days  ? 

MODEL.  15  is  3  times  6.  3  times  the  number  of  men  can  do 
the  work  in  \  of  the  time.  ^  of  9  days  is  3  days.  Ann.  3  days. 

2.  If  2  pipes  will  empty  a  cistern  in  1^  hours,  how 
many  such  pipes  will  empty  it  in  10  rain.  ? 

3.  If  2  barrels  of  flour  will  last  8  persons  6f  weeks, 
how  many  days  will  it  last  4  persons  ? 

4.  If  a  person  walks  %  of  a  mile  in  12  minutes,  how 
far  at  that  rate  will  he  walk  in  an  hour?    How  far  in 
1  hour  36  minutes  ? 

5.  If  8  yards  of  muslin  cost  $1.92,  how  much  will 
24  yards  cost  ? 


MISCELLANEOUS   EXAMPLES.  103 

6.  How  long  will  it  take  a  person  working  8  hours 
a  clay  to  paint  a  house,  if  he  can  do  it  in  10  clays,, 
working  12  hours  a  day  ? 

7.  If  3  tons  of  hay  cost  $57,  what  cost  4-J-  tons? 

8.  How  far  will  a  locomotive,  moving  at  the  rate 
of  6  miles  in  15  minutes,  go  in  an  hour  and  a  half? 

9.  If  a  barrel  of  flour  lasts  6  adults  and  3  children 
20  days,  how  long  at  the  same  rate  will  it  last  15  adults, 
rating  2  children  as  equal  to  1  adult  ? 

10.  What  is  the  freight  on  7  tons  4  cwt.,  at  the 
rate  of  $4.50  a  ton? 

11.  If  4  loads  of  hay  will  serve  6  horses  3  weeks, 
how  many  weeks  will  5  such  loads  serve  9  horses  ? 

MODEL.  If  4  loads  serve  6  horses  3  weeks,  1  load  will  serve  6 
horses  J  of  3  weeks,  or  f  of  a  week, — and  will  serve  1  horse  6  times 
f  of  a  week,  or  4£  weeks.  Five  loads  will  therefore  serve  1  horse  5 
times  4£,  or  42-,  weeks, — and  will  serve  9  horses  ^  of  \~  weeks,  or 
f ,  or  2|  weeks.  Ans.  2£  weeks. 

12.  How  many  acres  can  6  men  mow  in  10  days,  if 
4  men  can  mow  30  acres  in  5  days  ? 

13.  If  3  men  cut  12  cords  of  wood  in  6  days,  how 
many  days  will  it  take  4  men  to  cut  9  cords? 

14.  If  it  takes  12  men  5  days  to  do  a  certain  piece 
of  work,  how  many  men  will  it  take  to  do  three  times 
as  much  work  in  10  days  ? 

15.  If  the  freight  on  4  tons  of  merchandise  for  20 

o 

miles  is  $6,  how  much  is  it,  at  the  same  rate,  on  15 
cwt.,  for  30  miles  ? 

16.  If  6  men  in  4  days  can  build  80  rods  of  stone 
wall,  how  many  men  will  be  required  to  extend  the 
same  wall  a  mile  further  in  12  days  ? 


104  MISCELLANEOUS   EXAMPLES. 

SECTION  62. — 1.  Three  fifths  of  the  persons  at  a 
certain  meeting  were  ladies.  If  there  were  96  gentle- 
men present,  how  many  ladies  were  at  the  meeting  ? 

2.  If  -|  of  §120  is  4  times  the  cost  of  my  coat,  and 
ray  vest  costs  ^  as  much  as  my  coat,  what  is  the  cost 
of  both  coat  and  vest  ? 

3.  If  from  a  certain  number  increased  by  7  you 
subtract  3,  and  multiply  the  remainder  by  J  of  20.  the 
product  is  80  ;  wThat  is  the  number  ? 

4.  P  is  worth  $3600  ;  and  f  of  this  sum  is  twice  the 
value  of  Q's  property.     How  much  is  Q  worth  ? 

5.  -|  of  Oscar's  age  is  8  years  less  than  f  of  Lucy's, 
and  in  5  years  Lucy  will  be  32  ;  how  old  is  Oscar? 

6.  -J-  of  the  men  in  a  regiment  were  killed,  -J  wound- 
ed, and  £  captured;  300  escaped  uninjured.     Of  how 
many  was  the  regiment  composed? 

7.  F's  property  consists  of  a  house,  land,  and  stock. 
His  house  is  worth  f  of  the  whole,  his  stock  %,  and  his 
land  is  worth  $400.      What  is  his  whole  property 
worth  ?     What  is  his  house  worth  ?     His  stock  ? 

8.  James  is  18  years  old,  and  -|  of  his  age  is  |  of 
half  his  brother's  age  ;  how  old  is  his  brother  ? 

9.  A  is  three  score  years  and  ten.     If  ^  of  B's  age 
is  i  of  C's,  and  £  of  GTs  is  TV  of  A's,  how  old  is  B  ? 

10.  If  to  -|  of  the  number  of  sheep  in  a  certain 
flock  you  add  70,  you  will  double  their  number ;  how 
many  sheep  are  in  the  flock  ? 

MODEL.  70  sheep  equal  the  difference  between  f  of  the  flock 
and  twice  the  flock,  or  *£-.  15a— £=£.  If  70  sheep  are  seven  fifths 
of  the  flock,  one  fifth  is  |  of  70,  or  10;  and/ve  fifths,  or  the  whole 
flock,  are  5  times  10,  or  50.  Ans.  50  sheep. 


MISCELLANEOUS    EXAMPLES.  105 

11.  John,  having  lost  all  but  f  of  his  marbles,  won 
48  more,  and  by  so  doing  doubled  his  original  num- 
ber.    How  many  had  he  at  first  ? 

12.  A  person,  having  lost  -f  of  his  chickens,  bought 
76  more,  and  then  found  that  he  had  three  times  as 
many  as  at  first ;  how  many  was  that  ? 

13.  Grace  is  now  £  the  age  of  Blanch  ;  were  she 
5  years  older,  she  would  be  half  Blanch's  age.    What 
is  the  age  of  each  ? 

14.  Hugh  spent  f  of  his  money  for  a  melon,  and 
the  rest  for  cherries.     If  the  melon  cost  20c.,  and  the 
cherries  were  6  for  a  cent,  how  many  cherries  did  he 
buy? 

1 5.  From  a  liberty-pole  24  feet  high,  -|  of  the  whole, 
less  5  feet,  was  sawed  off;  how  many  feet  were  left 
standing  ? 


SECTION  63,— 1.  A  pole  increased  by  \  of  its  own 

length  would  be  12  ft.  long ;  what  is  its  length  ? 

MODEL.  The  length  of  the  pole,  being  £  of  itself,  when  increased 
by  £  of  itself,  must  be  f  of  itself,  and  this  we  are  told  equals  1 2  ft. 
If  six  fifths  of  the  length  are  12  ft.,  one  fifth  is  -fc  of  12  ft.,  or  2  ft. ; 
and/w  fifths,  or  the  whole  length,  are  5  times  2  ft.,  or  10  ft.  Ans. 
10ft. 

2.  A  horse  was  sold  at  a  profit  of  ^  of  its  cost.     It 
brought  $112  ;  what  was  the  cost  ? 

3.  A  horse  was  sold  for  |-  of  its  cost,  and  thereby 
a  loss  of  $16  was  incurred ;  what  was  it  sold  for  ? 

4.  A  horse  was  sold  for  2£  times  its  cost,  and  a 
profit  of  $125  was  thereby  realized.     What  did  the 
horse  cost  ? 


106  MISCELLANEOUS    EXAMPLES. 

5.  When  to  a  lot  of  iron  are  added  two  other  lots, 
\  and  f  as  heavy  as  the  first,  the  whole  weighs  106  Ib. 
What  was  the  weight  of  the  first  lot  ? 

6.  Twice  A's  age,  increased  by  |  of  his  age,  is  56 
years  ;  how  old  is  A  ? 

7.  After  losing  8200  and  giving  away  8450,  a  per- 
son found  that  he  had  -|  of  his  property  left.     How 
much  was  his  property  worth  ? 

8.  One  tenth  of  a  fanner's  sheep  died,  and  f  were 
sold.     On  buying  as  many  more  as  then  remained,  he 
had  28  ;  what  number  had  he  originally  ? 

9.  After  selling  f  of  his  coal,  and  then  f  of  the  re- 
mainder, a  coal-dealer  found  that  he  had  20  tons  left. 
How  many  tons  had  he  at  first  ? 

10.  A  library  having  taken  fire,  f-  of  the  books 
were  burned,  |-  of  what  remained  stolen,  and  only  750 
saved.     How  many  books  were  in  the  library  ? 

11.  A  farmer  sent  f  of  his  grain  to  market.     \  of 
the  grain  sent  was  corn  ;  the  rest,  396  bushels,  was 
wheat.     How  much  grain  had  he  left  ? 

12.  |  of  the  cost  of  some  furniture,  increased  by 
$50,  equals  the  selling  price.     If  there  was  a  profit  of 
810,  what  did  the  furniture  cost? 

13.  If  from  ^  of  a  certain  number  you  subtract  6, 
the  remainder  is  50  ;  what  is  the  number? 

14.  If  to  j\  of  a  certain  number  you  add  4J-,  the 
sum  is  54  ;  what  is  the  number  ? 

15.  What  number  is  that,  |  of  which  increased  by 
IJis  27|? 

16.  What  number  is  that,  f  of  which  divided  by  6 
is  31? 


MISCELLANEOUS    EXAMPLES.  107 

SECTION  64. — Fractions  having  a  common  de- 
nominator are  to  each  other  as  their  numerators,  f  is 
to  f  as  4  is  to  5. 

Fractions  that  have  not  a  common  denominator 
may  be  reduced  to  other  fractions  that  have,  and  are 
to  each  other  as  the  numerators  of  the  latter.  |-  is  to 
f  as  8  is  to  9,  since  !=•&  and  f =TV 

1.  |  is  to  -J  as  what  two  numbers  ? 

2.  f  is  to  f  as  what  two  numbers  ? 

3.  f  is  to  f  as  what  two  numbers  ? 

4.  -^  is  to  ^  as  what  two  numbers  ? 

5.  £  is  to  |  as  what  two  numbers  ? 

6.  If  an  estate  is  divided  into  two  shares  that  are 
to  each  other  as  4  to  3,  what  parts  of  the  estate  will 
rhese  shares  be  ? 

MODEL.  To  find  shares  that  are  to  each  other  as  4  to  3,  we  di- 
vide the  whole  into  4  +  3  equal  parts  (that  is,  into  7  equal  parts,  or 
seventlis),  and  take  four  of  these  (-})  for  the  first,  and  five  ($•)  for  the 
second.  Ans.  %  and  f- . 

7.  If  we  divide  a  number  into  two  parts  that  are 
to  each  other  as  3  to  7,  what  fractions  of  the  number 
are  these  parts  ? 

8.  Three  partners,  A,  B,  and  C,  a^ree  that  A  shall 
have  $3  of  their  profits  to  B's  $2  and  C's  $1.     What 
part  of  the  profits  must  each  receive  ? 

9.  Suppose  that   these  partners  make  $600,  how 
many  dollars  should  each  receive? 

10.  To  divide  a  number  into  parts  that  are  to  each 
other  as  6  to  9,  what  fractions  of  it  must  we  take  ? 

11.  Divide  45  into  two  parts  that  are  to  each  other 
as  6  to  9. 


108  MISCELLANEOUS    EXAMPLES. 

12.  To  divide  a  number  into  three  parts  that  are  to 
each  other  as  2,  3,  and  4,  what  fractions  of  the  number 
must  we  take  ? 

13.  Divide  72  into  three  parts   that  are  to  each 
other  as  2,  3,  and  4. 

14.  To  divide  a  number  into  parts  that  are  to  each 
other  as  f  and  f,  what  fractions  of  it  must  we  take  ? 

15.  Divide  19  into  two  parts  that  shall  be  to  each 
as  f  and  f . 

16.  Two  numbers,  making  up  28,  are  to  each  other 
as  5  and  9  ;  what  are  the  numbers  ? 

17.  Divide  60  into  three  parts  that  shall  be  to  each 
other  as  2,  5,  and  3. 

18.  A  and  B  enter  into  a  speculation,  A  contribut- 
ing $200  and  B  $300.     The  profit  is  $140  ;  divide  it 
between  them  in  proportion  to  the  sums  contributed. 

19.  Two  boats,  leaving  places  100  miles  apart,  sail 
toward  each  other,  one  at  the  rate  of  12  miles  an  hour, 
and  the  other  8.     By  the  time  they  meet,  how  many 
miles  has  each  gone  ? 

20.  Three  partners  divide  their  profits  according  to 
the  money  they  put  in.     The  first  put  in  $1000,  the 
second  $3000,  the  third  $4000.     Their  profit  for  a  year 
being  $2800,  how  much  should  each  have  ? 

21.  A  man  has  $500  to  divide  between  three  cred- 
itors, to  whom  he  owes  respectively  $1000,  $1500,  and 
$2500.     They  are  to  be  paid  in  proportion  to  their 
claims  ;  how  much  should  each  receive  ? 

22.  A,  B,  C,  and  D,  hire  a  pasture  for  $18.     A  turns 
in  4  cows,  B  6,  C  3,  and  D  5 ;  how  much  should  each 
pay? 


PARTNERSHIP.  109 

SECTION  65.— -1.  A  and  B  make  a  purchase  in 
partnership.  A  contributes  $300  for  2  months,  and  B 
$400  for  3  months.  Their  profit  is  $450  ;  how  should 
it  be  divided  between  them  ? 

MODEL.  A  puts  in  $300  for  2  months,  which  is  equivalent  to 
$600  for  1  month.  B  puts  in  $400  for  3  months,  which  is  equiva- 
lent to  $1200  for  1  month.  The  profit  should  therefore  be  divided 
between  them  in  the  proportion  of  600  to  1200,  or  1  to  2.  A  should 
have  ^  of  $450,  or  $150 ;  and  B,  f  of  $450,  or  $300.  Ans.  A, 
$150;  B,  i 


2.  Two  parties  received  $165  for  digging  a  drain. 
How  should  it  be  divided  between  them,  if  the  first 
furnished  6  laborers  for  5  days,  and  the  second  12  la- 
borers for  3  days  ? 

3.  A  pasture  is  hired  by  two  persons  for  $16.     The 
first  turns  in  10  cows  for  two  months,  the  second  2 
cows  for  six  months  ;  how  much  should  each  pay  ? 

4.  Three  persons  hired  a  pasture  for  $30.    The  first 
turned  in  7  cows  for  5  months,  the  second  5  cows  for 
4  months,  and  the  third  25  sheep  for  4^  months.    Reck- 
oning the  pasturage  of  5  sheep  worth  that  of  2  cows, 
how  much  ought  each  to  pay  ? 

5.  Two  parties  contract  to  do  some  mowing  for 
$38.50.     The  first  works,  with  his  three  boys,  4  days  ; 
the  second  furnishes  three  men,  and  works  with  them 
himself,  3  days.     Reckoning  2  boys  as  equal  to  1  man, 
how  should  the  pay  be  divided  ? 

6.  A  and  B  gave  $20.50  for  a  ton  of  hay,  to  be  di- 
vided between  them.     If  A  paid  $8.20  of  the  purchase 
money,  how  many  hundred-weight  should  he  take,  and 

hpw  many  B  ? 

10 


110  MISCELLANEOUS    EXAMPLES. 

SECTION  66. — 1.  If  an  orange  costs  twice  as  much 
as  a  lemon,  and  together  they  cost  6  cents,  how  much 
does  each  cost  ? 

MODEL.  The  cost  of  the  lemon  is  once  itself;  the  cost  of  th^ 
orange  is  twice  that  of  the  lemon.  Then  the  cost  of  both  (given  at 
6  cents)  must  be  three  times  that  of  the  lemon.  If  6c.  is  three 
times  the  cost  of  the  lemon,  the  cost  of  the  lemon  must  be  •£  of  6c.. 
or  2c. ;  and  that  of  the  orange  must  be  twice  2c.,  or  4c.  Ans.  The 
lemon,  2c. ;  the  orange,  4c. 

2.  Robert's  age  is  3  times  Helen's,  and  together 
they  are  36.     How  old  is  each  ? 

3.  A  farmer  has  4  times  as  many  sheep  as  cows,  and 
•of  both  he  has  55  ;  how  many  sheep  has  he  ? 

4.  Charles  and  Jane  together  have  29  books.     If 
'Charles  has  7  more  than  Jane,  how  many  has  each? 

5.  Two  boys  have  ^  of  $1  between   them;  how 
many  cents  has  each,  if  the  first  has  12  cents  less  than 
the  second  ? 

6.  Ida  has  4  more  roses  than  twice  as  many  as 
Dora  ;  together  they  have  32  ;  how  many  has  each  ? 

7.  A  man  gave  $350  for  a  horse,  wagon,  and  har- 
ness.    He  gave  3  times  as  much  for  the  wagon  as  for 
the  harness,  and  twice  as  much  for  the  horse  as  for  the 
wagon  ;  what  was  the  cost  of  each  ? 

8.  A  and  B  start  with  equal  sums  of  money.     A 
:  gains  $45,  B  loses  $30  ;  and  together  they  then  have 

$115.     How  much  had  each  at  first  ? 

MODEL.  If  A  gains  $45  and  B  loses  $30,  they  must  then  have 
had  845—$30,  or  $15,  more  than  they  had  at  first.  If  $115  is  $15 
more  than  they  had  at  first,  they  must  together  have  had  $115— 
•  $15,  or  8100 ;  and  each  must  have  had  £  of  $100,  or  $50.  Am.  $50. 


MISCELLANEOUS    EXAMPLES.  Ill 

9.  A  counter,  show-case,  and  desk,  cost  $69.     The 
counter  cost  $7  less  than  the  show-case,  and  the  desk 
$8  more  than  the  counter ;  what  did  each  cost  ? 

10.  Sold  a  ring,  watch,  and  breast-pin,  for  $110. 
The  watch  brought  5  times  as  much  as  the  ring,  and 
the  pin  $5  more  than  the  ring;  what  did  each  bring  ? 

11.  A  lady  bought  a  table,  easy-chair,  and  sofa,  for 
$88.     For  the  table  she  gave  $13  more  than  for  the 
chair,  and  for  the  sofa  $35  more  than  for  the  table. 
What  did  she  give  for  each  ? 


SECTION  67.— 1.  Two  pipes,  entering  a  vat,  will 
fill  it,  the  one  in  10  minutes,  the  other  in  15  minutes. 
What  part  of  the  vat  will  each  fill  in  1  minute  ?  What 
part  will  both  fill  in  1  minute  ?  How  many  minutes 
will  it  take  both  pipes  to  fill  the  vat  ? 

2.  A  man  can  do  a  certain  job  in  5  days,  and  a  boy 
in  10  days.     How  long  will  it  take  the  man  and  boy, 
working  together,  to  do  %  the  job  ?    To  do  £  the  job  ? 

3.  A  can  do  a  job  in  10  days,  and  B  in  12  clays. 
After  A  has  been  working  5  days,  how  long  would  it 
take  B  to  finish  it  ?    After  B  has  been  working  3  days, 
how  long  would  it  take  A  to  finish  it  ? 

4.  A  can  mow  a  certain  field  in  15  hours,  B  in  12 
hours.     After  they  have  been  mowing  2  hours,  they 
call  in  C,  who  could  do  it  alone  in  the  same  time  as  B. 
In  how  many  hours  will  the  three  finish  it  ? 

5.  Philip  can  split   some  wood  in   8  days ;  with 
Frank's  help  he  can  do  it  in  5  days.     In  how  many 
days  could  Frank  do  it  alone  ? 


112  MISCELLANEOUS    EXAMPLES. 

6.  F,  G,  and  H,  can  dig  a  certain  cellar  in  4  days ; 
F  and  H  can  do  it  in  6  days,  and  G  and  H  in  8  days. 
How  long  will  it  take  each  to  dig  it  ? 

MODEL.  If  all  three  can  dig  it  in  4  days,  in  1  day  they  can  dig 
£  of  it ;  and,  if  F  and  H  can  dig  it  in  6  days,  in  1  day  they  can  dig 
£  of  it.  Then  G  in  1  day  can  do  the  difference  between  ^  and  £,  or 
-jV ;  and  to  do  |f ,  or  the  whole,  will  take  G  as  many  days  as  •&  is 
contained  times  hi  |f,  or  12  days. 

Reasoning  in  the  same  way  with  G  and  H,  we  find  that  F  can  do 
£  of  it  in  1  day,  or  the  whole  in  8  days.  Now,  if  F  and  H  can  do  £ 
of  it  in  1  day,  and  F  alone  i  of  it  in  1  day,  H  can  do  in  1  day  the 
difference  between  £  and  -J-,  or  jL,  and  can  therefore  do  the  whole  in 
24  days.  Ans.  F  8  days,  G  12  days,  and  H  24  days. 

7.  Harvey,  Louis,  and  Oliver,  can  thrash  some  rye 
in  2£  days.     Without  Oliver's  help  they  can  do  it  in 
4  days  ;  how  long  would  it  take  Oliver  to  do  it  alone  ? 

8.  A  vat  is  emptied  by  3  pipes  in  1|  minutes.  The 
1st  and  3d  can  empty  it  in  2f  minutes,  and  the  2d  and 
3d  in  3  minutes.     In  what  time  will  each  empty  it  ? 
In  what  time  will  the  1st  and  2d  together  empty  it  ? 

9.  Three  casks,  A,  B,  and  C,  hold  105  gal.     A  and 
B  hold  65  gal.,  B  and  C  75  ;  how  much  does  each  hold  ? 

10.  Required  the   contents  of  three  casks,  if  the 
first  and  second  hold  42  gal.,  the  second  and  third  46 
gal.,  and  the  first  and  third  44  gal. 

NOTE.  As  the  contents  of  each  cask  are  taken  twice,  the  sum 
of  42,  46,  and  44  gal.  must  be  twice  the  contents  of  the  three  casks. 
After  thus  finding  the  contents  of  the  whole,  proceed  as  before. 

11.  Three  pipes  feed  a  reservoir.  The  first  and 
second  can  fill  it  in  3  days,  the  first  and  third  in  4 
days,  the  second  and  third  in  2|  clays.  How  long  will 
it  take  each  to  fill  it  ? 


MISCELLANEOUS    EXAMPLES.  113 

SECTION  68.— 1.  A  pole  3|  yd.  long  is  divided 
into  equal  parts  by  5  notches.  How  many  feet  and 
inches  are  there  in  each  division  ? 

2.  Two  men  travel  from  the  same  point,  in  the 
same  direction,  one  at  the  rate  of  3  mi.  1  fur.  an  hour, 
the  other  at  the  rate  of  2  mi.  7  fur.  20  rd.  an  hour. 
How  far  apart  will  they  be  in  half  a  day  ? 

3.  Two  men  travel  from  the  same  point,  in  oppo- 
site directions,  one  at  the  rate  of  4  mi.  6  fur.  an  hour, 
the  other  at  the  rate  of  5  mi.  5  fur.  an  hour.     How  far 
apart  will  they  be  in  30  minutes  ? 

4.  In  New  England,  6s.  make  a  dollar.     How  many 
dollars  will  40  yd.  of  calico  cost,  at  9d.  a  yard  ? 

MODEL.  6s.  (which  in  N.  E.  is  $1)  equals  72d. ;  9d.  is  there- 
fore -&,  or  £,  of  $1.  At  $1  a  yard,  40  yd.  would  cost  $40 ;  and  at  i 
of  a  dollar  a  yard  they  cost  \  of  $40,  or  $5.  Ans.  $5. 

5.  At  9s.  a  day,  how  many  dollars  will  it  cost  to 
hire  three  men  for  four  days  in  Massachusetts  ? 

6.  How  many  primers,  at  9d.  apiece,  can  be  bought 
for  $3.50  in  New  Hampshire  ? 

7.  If  Emma  can  pick  a  quart  of  berries  in  J  of  an 
hour,  and  Rose  can  pick  a  quart  in  T3F  of  an  hour,  how 
many  quarts  can  both  pick  in  2  h.  30  min.  ? 

8.  Two  pipes  can  till  a  vat  in  -J-  of  an  hour.     The^ 
first  can  fill  it  in  half  an  hour;  how  many  such  vats, 
could  the  secowd  fill  in  one  hour  ? 

9.  A  girl,  buying  some  paper,  needed  lOc.  more  to 
get  the  best,  which  was  2c.    a  sheet,  but  had  just 
enough  to  buy  what  was  sold  at  the  rate  of  2  sheette 
for  3c.     How  many  sheets  did  she  want  ? 

10.  Edward  bought  4  tops,  and  had  10  cents  left.. 


114  MISCELLANEOUS   EXAMPLES. 

Had  he  bought  7  such  tops,  he  would  have  had  but  Ic, 
left.     How  much  were  the  tops  apiece  ? 

11.  A  lady  gave  each  of  her  children  $3,  and  had 
89  left  in  her  purse.     Had  she  wanted  to  give  each  $5, 
she  would  have  needed  $5  more.     How  many  children 
had  she  ? 

12.  Horatio  is  14  years  older  than  Valorus,  and  to 
gether  they  are  41.     How  old  is  each  ? 

13.  The  difference  of  two  numbers  is  10,  their  sum 
is  28  ;  what  are  the  numbers  ? 

14.  What  two  numbers  are  those,  whose  sum  is  125, 
and  their  difference  25  ? 


SECTION  69,— 1.  A  hound,  running  after  a  fox, 
gains  20  rods  on  him  every  minute.  If  the  fox  has 
half  a  mile  the  start,  how  long  before  he  will  be 
caught  ? 

2.  A  deer  is  \  of  a  mile  before  a  hound,  and  every 
minute  the  deer  runs  60  rods,  and  the  hound  70.     How 
long  will  it  be  before  the  deer  is  overtaken  ? 

3.  A  policeman  runs  5  rods  to  a  thiefs  4.     The 
thief  has  10  rods  the  start ;  how  far  will  the  policeman 
run  before  he  catches  the  thief? 

MODEL.  If  the  policeman  runs  5  rods  to  the  thiefs  4,  he  gains 
1  rod  on  every  5  he  runs ;  and,  to  gain  10  rods,  he  will  have  to  rua 
as  many  times  5  rods  as  1  rod  is  contained  times  in  10  rods,  or  10. 
U>  times  5  is  50.  Am.  50  rods. 

4.  How  far  will  the  thief  mentioned  in  the  last 
Example  run  before  he  is  caught  ? 


MISCELLANEOUS   EXAMPLES.  115 

5.  A  hare  is  25  leaps  in  advance  of  a  hound,  and 
takes  4  leaps  to  the  hound's  3,  but  1  leap  of  the  hound 
equals  1£  leaps  of  the  hare.     How  many  leaps  will 
each  take  before  the  hare  is  caught  ? 

6.  A  train  of  cars  is  10  miles  behind  a  stage.     If 
the  cars  go  5  times  as  fast  as  the  stage,  how  many 
miles,  &c.,  will  the  stage  go  before  it  is  overtaken  ? 

7.  At  what  time  between  2  and  3  o'clock  are  the 
minute  and  hour  hand  of  a  watch  together? 

MODEL.  The  minute  and  hour  hand  are  together  at  12  o'clock. 
In  the  course  of  12  hours,  the  minute  hand  will  overtake  the  hour 
hand  11  times;  to  overtake  it  once,  therefore,  will  require  ft  of  12 
hours,  or  l-ft  hours. 

When  the  hands  are  together  between  2  and  3,  the  minute  hand 
will  have  overtaken  the  hour  hand  twice  since  1 2  o'clock,  which  will 
require  twice  1ft  hours,  or  2ft  hours.  Ans.  2ft  hours  past  12,  or 
10  min.  54ft  sec.  past  2. 

8.  At  what  time  between  3  and  4  will  the  hour  and 
minute  hand  of  a  clock  be  together  ? 

9.  At  what  time  between  7  and  8  will  the  hands  be 
together  ?    At  what  time  between  9  and  10  ? 

10.  A    person   received   from   his   employer  $74, 
under  an  agreement  that  he  was  to  board  with  the  lat- 
ter 40  days,  and  receive  $2.50  a  day  when  he  worked, 
and  lose  75c.  a  day  when  he  was  idle.     How  many 
days  was  he  idle  ? 

MODEL.  His  pay  being  $2.50  a  day,  if  he  had  worked  the  whole 
40  days,  he  would  have  received  40  times  $2.50,  or  $100 ;  hence  he 
lost  by  being  idle  the  difference  between  $100  and  $74,  or  $26.  Each 
day  he  was  idle  he  failed  to  make  $2.50,  and  forfeited  75c.  besides, 
thus  losing  in  all  $3£.  Hence,  if  he  lost  $26,  he  must  have  been  idle 
as  many  days  as  $3£  is  contained  times  in  $26,  or  8.  Am.  8  days. 


116  MISCELLANEOUS    EXAMPLES. 

SECTION  70. — 1.  A  contracted  to  work  36  days 
for  B,  on  condition  that  he  should  receive  $2  for  every 
day  he  worked,  and  forfeit  50c.  each  day  he  was  idle. 
If  he  received  from  B  $47,  how  many  days  was  he  idle  ? 

2.  How  many  square  yards  in  an  oblong  surface  7 
yards  long  and  4  yards  wide? 

MODEL.  A  surface  1  yd.  long  and  1  yd.  wide  contains  1  sq.  yd. ; 
a  surface  7  yd.  long  and  1  yd.  wide,  would  therefore  contain  7  times 
1,  or  7,  sq.  yd. ;  and  a  surface  7  yd.  long  and  4  yd.  wide  contains  4 
times  7,  or  28,  sq.  yd.  Ans.  28  sq.  yd. 

3.  How  many  square  rods  in  an  oblong  field  20 
rods  by  15  rods  ? 

4.  How  many  sq.  yd.  in  a  floor  12  ft.  square? 

5.  At  30c.  a  square  yard,  what  will  it  cost  to  plas- 
ter a  wall  10  ft.  high  and  18  ft.  long  ? 

6.  What  will  it  cost  to  paint  a  wall  9  yd.  long  and 
4  yd.  high,  at  5c.  per  square  foot  ? 

7.  How  many  marble  blocks  12  in.  square  will  be 
needed  to  pave  a  walk  3  ft.  wide  and  20  ft.  long  ? 

8.  How  much  yard-wide  carpeting  will  be  required 
to  cover  a  room  18  ft.  by  15  ft.  ? 

9.  How  much  carpeting  f  of  a  yard  wide  is  required 
for  a  room  9  yd.  long  by  14  ft.  wide? 

10.  How  many  cubic  feet  in  a  box  4  ft.  long,  3  ft. 
wide,  and  2  ft.  high  ? 

MODEL.  A  box  1  ft.  long,  1  ft.  wide,  and  1  ft.  high,  would  con- 
tain 1  cu.  ft. ;  a  box,  therefore,  4  ft.  long,  1  ft.  wide,  and  1  ft.  high, 
would  contain  4  times  1,  or  4,  cu.  ft.;  a  box  4  ft.  long,  3  ft.  wide, 
and  1  ft.  high,  would  contain  3  times  4,  or  12,  cu.  ft. ;  and  a  box  4 
ft.  long,  3  ft.  wide,  and  2  ft.  high,  contains  twice  12,  or  24,  cu.  ft. 
Ans.  24  cu.  ft. 


PERCENTAGE.  11 7 

11.  How  many  cubic  yards  in  a  bin  5  yd.  long,  2 
yd.  wide,  and  3  yd.  high  ? 

12.  What  is  the  freight  on  a  box  3  ft.  long,  3  ft. 
wide,  and  3  ft.  high,  at  50c.  a  .cubic  foot  ? 

18.  What  will  it  cost  to  dig  a  cellar  12  feet  long, 
wide,  and  high,  at  60c.  a  cubic  yard  ? 


CHAPTER     ELEVENTH. 

PERCENTAGE. 

SECTION  71.— 1.  What  do  the  words  Per  cent, 
mean  ?    Ans.  By  or  on  the  hundred. 

2.  What  does  1  Per  cent,  mean,  and  how  may  it  be 
written  ?    Ans.  I  Per  cent,  means  one  on  every  hun- 
dred^ or  1  hundredth,  and  may  be  written  1  %. 

3.  What  does  2  Per  cent,  mean,  and  how  may  it  be 
written  ?     Ans.    2  Per  cent,  means  two  hundredths 
(y|¥),  and  may  be  written  2  <fc. 

4.  What  part  is  3  %  ?     7  fc  ?     11  %  ?     13  %  ? 

5.  What  part  is  50  %  ?    Ans.    -ffo,  or  \. 

6.  What  part  is  25  %  ?     What  part  is  20  %  ? 

7.  What  part  is  10  %  ?     What  part  is  5  tf  ? 

8.  What  part  is  4  %  ?     What  part  is  2  #  ? 

9.  What  part  is  6  %  ?     What  part  is  40  %  ? 

10.  What  part  is  80  %  ?     What  part  is  35  %  ? 

11.  What  per  cent,  is  \  equal  to  ? 

MODEL.     A  whole  is  100  %  of  itself,  and  \  is  £  of  100  #,  or  60  #. 
4w«.  50$. 


118  PERCENTAGE. 


12.  What  per  cent,  is  £  equal  to  ?     {-  ?     T^  ? 

13.  What  per  cent,  is  ^  equal  to  ?    ±  ?     T^  ? 

14.  What  per  cent,  is  £  equal  to  ?     -^  ?     ^  ? 

15.  What  per  cent,  is  f  equal  to  ?     |  ?     ^  ? 

16.  What  per  cent,  is  f  equal  to  ?     J  ?     ^  ? 
IV.  What  part  is  33J  #  ?     What  part  is  66|  # 
18.  What  part  is  16-|  %  ?     What  part  is  12J-  # 


SECTION  72.— 1.  If  a  person  loses  f  of  his  prop- 
erty, what  per  cent,  of  it  has  he  left  ? 

2.  If  a  person  sells  30  f0  of  his  sheep,  what  part  of 
his  flock  does  he  retain  ? 

3.  How  much  is  5  $  of  $500  ? 

MODEL.  5  £=TiTo-  rio  of  $500  is  $5,  and  T  J-ff  is  5  times  $5,  or 
$25.  Ans.  $25. 

4.  How  much  is  4  #  of  $25  ? 

MODEL.    4  £=Tifl,  or  &.     &  of  $25  is  $1.     Ans.  $1. 

XOTE.  It  is  sometimes  best  to  reduce  the  fraction  to  its  lowest 
terms,  as  in  this  Example, — and  sometimes  not,  as  in  the  last.  The 
pupil  must  exercise  his  judgment. 

5.  How  much  is  3  %  of  $175  ?  10  %  of  $345  ? 

6.  How  much  is  6  ^  of  $150  ?  25  %  of  $960  ? 

7.  How  much  is  9  %  of  $400  ?  75  %  of  $3.60  ? 

8.  How  much  is  4  %  of  $2.50  ?  60  #  of  $4.75  ? 

9.  How  much  is  8  %  of  $9.00  ?  50  #  of  $8.88  ? 

10.  How  much  is  33J#  of  £36  ?  12^^  of  8  bu.  ? 

11.  How  much  is  20  %  of  1  mi.  ?  16|  #  of  £4  ? 

12.  How  much  is  8J  #  of  27  A.  ?  40  %  of  $75  ? 

13.  How  much  is  7  #  of  $1000  ?  11  #  of  25  gal.  ? 


PERCENTAGE.  119 

14.  How  much  is  2  #  of  $20  ?     16  %  of  12J  ft.  ? 

15.  The  profit  on  certain  goods,  bought  for  $120, 
was  33 1  %  ;  how  many  dollars  was  it  ? 

16.  What  must  goods  bought  for  $120  be  sold  for, 
in  order  to  make  33£$  ? 

17.  What  was  the  profit  on  goods  bought  for  $150, 
and  sold  at  an  advance  of  20  $  ? 

18.  What  was  the  loss  on  a  farm  bought  for  $1200, 
and  sold  at  5  %  below  cost  ?     What  did  it  bring  ? 

19.  A  merchant,  having  bought  1  dozen  pair  of 
boots  for  $72,  wishes  to  make  16f$  on  them.     How 
much  must  he  charge  a  pair  ? 

20.  A  and  B  had  $200  each.     A  gained  6  %  on  his 
capital ;  B  lost  12|  %  on  his.     How  much  more  was  A 
then  worth  than  B  ? 

21.  A  farmer,  having  80  chickens,  lost  25  </0  of  them, 
and  then  sold  75  %  of  the  remainder  ;  how  many  chick- 
ens had  he  left  ? 

22.  A  collector,  who  charges  5  <f0  commission,  col- 
lects two  bills,  one  of  $30,  the  other  of  $150;  how 
much  should  he  retain,  and  how  much  pay  over  ? 

23.  Bought  some  goods  for  $600  ;  sold  20  %  of  them 
at  a  loss  of  10$,  arid  the  rest  at  an  advance  of  5$. 
Did  I  gain  or  lose  on  the  whole,  and  how  much  ? 

24.  A  hogshead  of  molasses,  containing  60  gal., 
was  bought  for  $30.    50  <fc  of  the  contents  having  leaked 
out,  what  must  the  rest  be  sold  for  a  gallon,  so  as  not 
to  lose  on  the  whole  ? 

25.  How  much  is  100  %  of  999  ?     What  per  cent,  is 
anything  of  itself? 

26.  How  much  is  200  %  of  47  ?     300  %  of  56  ? 


120  PERCENTAGE. 

SECTION  73.— 1.  What  f0  of  $500  is  §10  ? 

MODEL.     $10  is  ^ft,,  or  ^r,  of  $500.     ^,=2  %.    Am.  2  %. 

2.  What  %  is  $5  of  $50  ?     £6  of  £150  ? 

3.  What  %  is  $9  of  $36  ?     $3.30  of  $16.50  ? 

4.  1  cwt.  of  1  ton  (20  cwt.)  ?     2  dimes  of  $1  ? 

5.  1  foot  of  1  yard  ?     3  pecks  of  a  bushel  ? 

6.  A  man  bought  some  goods  for  $100,  and  sold 
them  at  a  loss  of  $7  ;  what  was  the  rate  %  of  loss  ? 

7.  What  was  the  rate  of  loss  on  goods  bought  for 
$100,  and  sold  for  $93  ? 

NOTE.    The  rate  of  gain  or  loss  must  always  be  reckoned  on  the 
cost. 

8.  Some  muslin,  bought  for  $96,  was  sold  for  $102 ; 
what  was  the  rate  of  profit  ?    What  would  it  have 
been,  had  the  muslin  brought  $104  ? 

9.  A  bookseller  lets  a  teacher  have  a  dollar  book 
for  75c. ;  what  %  does  he  take  off? 

10.  A  house  was  put  up  for  $3000,  on  a  lot  that 
cost  $600.     The  whole  was  sold  for  $4500.     Did  the 
owner  gain  or  lose,  and  what  rate  %  ? 

11.  An  agent  gets  $9  for  collecting  a  bill  of  $300 ; 
what  %  does  he  receive  ? 

12.  If  a  house  rents  this  year  for  $450,  which  last 
year  brought  $25  a  month,  how  much  per  cent,  has  the 
rent  advanced  ? 

13.  80  is  33|  %  of  what  number? 

MODEL.     33^=^.     If  80  is  \  of  the  required  number,  J,  or  the 
whole,  must  be  3  times  80,  or  240.     Am.  240. 

14.  16  is  40  #  of  what  number  ? 

15.  15  Ib.  is  3  %  of  how  many  cwt.  ? 


PERCENTAGE.  121 

16.  50  %  of  $50  is  25  %  of  what  ? 

17.  A  collector,  who  charges  10$,  receives  $40  for 
collecting  a  bill ;  what  was  its  amount  ? 

18.  Selling  some  goods  at  7$  advance,  a  person 
makes  $14  ;  what  did  the  goods  cost  ? 

19.  A  planter  lost  $21  on  a  horse;  if  his  loss  was 
at  the  rate  of  14  $,  what  was  the  cost  of  the  horse  ? 
What  did  he  sell  the  horse  for  ? 

20.  Selling  a  house  for  6  %  less  than  it  cost,  a  per- 
son  lost  $120  ;  what  was  the  selling  price  ? 

21.  20  %  of  £10  is  6  %  of  what  ? 


SECTION  74.— 1.  After  gaining  50$  of  his  capital, 
a  trader  had  $1800  ;  what  was  his  capital  ? 

MODEL.  His  capital  was  100  %  of  itself.  After  gaining  50  %  of 
his  capital  more,  he  had  150$,  or  f,  of  his  capital.  If  $1800  is  f- 
of  his  capital,  \  of  his  capital  is  \  of  $1800,  or  $600 ;  and  f-,  or  the 
whole  capital,  is  twice  $600,  or  $1200.  Ans.  $1200. 

2.  A  person  gave  $160  for  an  ox,  which  was  33|$ 
more  than  its  real  value ;  what  was  its  value  ? 

3.  If  a  lot,  sold  for  $495,  brings  10  %  less  than  its 
value,  at  what  price  would  it  have  brought  10  $  more 
than  its  value  ? 

4.  A  certain  number,  diminished  by  12  J$  of  itself, 
is  35  ;  what  is  the  number  ? 

5.  A  man  gave  his  daughter  25  <f0  of  the  rent  of  a 
certain  house,  and  the  house  rented  for  10$  of  its 
value.     If  the  daughter  received  $50,  how  much  was 
the  house  worth  ? 

11 


122  PERCENTAGE. 

6.  P  sold  a  horse  to  Q  at  a  profit  of  5  % ;  Q  sold 
him  to  R  at  a  profit  of  10$.     It*  R  paid  $231  for  the 
horse,  what  did  P  pay  ? 

7.  A  person  sent  his  agent  $840,  to  pay  for  an  in- 
vestment and  the  agent's  charge  of  5  %  on  the  same. 
What  was  the  amount  of  the  investment  ? 

8.  An  agent,  having  collected  some  money,  retained 
his  commission,  which  was  at  the  rate  of  2  $,  and  paid 
over  $147.     What  was  the  amount  collected,  and  how 
much  was  his  commission  ? 

9.  A  collected  for  B  20  bills  of  equal  amount,  and, 
after  deducting  10  %  for  his  commission,  paid  over  the 
balance,  $108.     How  much  was  each  bill  ? 

10.  In  a  grove  of  pine,  spruce,  and  cedar  trees, 
25  <fa  of  the  trees  are  pines,  and  35  $  spruces.     If  there 
are  32  cedars,  how  many  trees  does  the  grove  contain  ? 

1 1.  Fifteen  per  cent,  of  my  peach  trees  having  died, 
I  set  out  39  more,  and  then  had  50  %  more  than  at  first. 
How  many  had  I  at  first  ? 

12.  Having  gained  20$  on  his  capital,  a  merchant 
lost  $160,  and  found  that  he  then  had  in  all  $1400. 
What  was  his  original  capital  ? 

13.  A  person  sold  6  watches  for  $60  apiece,  which 
was  25  %  less  than  they  cost  ?    How  much  did  he  lose 
on  the  whole  ? 

14.  To-day  I  take  in  $20,  which  is  20  <£  less  than  I 
took  yesterday,  and  33 \  %  more  than  I  took  the  day 
before.     What  are  my  receipts  for  all  three  days  ? 

15.  Having  doubled  his  money,  B  gave  awray  7  $ 
of  what  he  then  had,  and  found  that  he  had  $279  left. 
How  much  had  he  at  first  ? 


BANKRUPTCY.  123 

SECTION  75.— 1.  What  is  a  Bankrupt  ?  Am.  One 
who  fails  in  business  and  can  not  pay  his  debts. 

2.  What  is  meant  by  a  bankrupt's  Assets?    Ans. 
The  property  in  his  hands. 

3.  What   is   meant   by  a  bankrupt's  Liabilities? 
Ans.  His  debts  or  obligations. 

4.  If  a  bankrupt  can  pay  25  cents  on  the  dollar, 
what  per  cent,  of  his  debts  can  he  pay  ?    How  much 
should  A  receive,  whom  he  owes  $200  ? 

5.  A  fails,  owing  $9000,  and  having  $1500  assets. 
What  </0  will  his  creditors  get  on  their  claims  ? 

MODEL.  As  A's  debts  are  $9000,  and  he  has  $1500  to  pay  them 
with,  each  dollar  of  debt  must  draw  ^-(]  of  $1500,  or  £ggg  of  $1. 
i3-8S=i  which  is  16  J  %.  Am.  16 J  %. 

6.  If  the  bankrupt  mentioned  in  the  last  Example 
owes  C  $420,  how  much  of  the  assets  should  C  receive  ? 

7.  A  person  fails,  having  four  creditors ;  he  owes 
A  $1200,  B  $1500,  C  $300,  and  D  $1000.     His  assets 
are  $500.     How  many  cents  can  he  pay  on  the  dollar, 
and  how  much  should  each  creditor  receive  ? 

8.  A  bankrupt  pays  75c.  on  the  dollar ;  how  much 
will  a  creditor  lose,  whom  he  owes  $920  ?    How  much 
will  a  creditor  receive,  whom  he  owes  $840  ? 

9.  If  a  creditor  receives  $420  for  a  debt  owed  him 
by  a  bankrupt  who  can  pay  but  40c.  on  the  dollar, 
how  much  was  the  debt  ? 

10.  A  person  fails.     His  liabilities  are  $8000  owed 
to  B,  and  20  %  of  that  amount  owed  to  C.     His  assets 
are  $400  cash,  $1600  in  goods,  and  $1200  in  notes. 
What  per  cent,  can  he  pay,  and  how  much  should  each 
of  his  creditors  receive  ? 


124  INSURANCE. 

SECTION  76.-— 1.  What  is  Insurance?  Ana.  Insur- 
ance is  a  contract,  by  which,  for  a  certain  sum  paid, 
one  party  secures  another  against  loss  by  fire,  the  clan- 
gers of  navigation,  &c. 

2.  What   is   the   sum  paid    called?      Ans.    The 
Premium. 

3.  A  paint-store  is  insured  to  the  amount  of  §5000, 
at  1  \  fc.     What  is  the  premium  ? 

MODEL.  At  1  #,  the  premium  would  be  ^  of  f  5000,  or  $50 ;  at 
$  £,  it  would  be  -£  of  $50,  or  £25.  $50  +  $25 =$75.  Ans.  $75. 

4.  A  house  is  insured  for  $2000,  and  the  furniture 
in  it  for  $1000.     What  does  the  insurance  cost,  at  J 
of  1  per  cent.  ? 

5.  Two  hundred  barrels  of  flour,  worth  $8  a  barrel, 
are  insured  for  75  %  of  their  value.     What  is  the  pre- 
mium, if  the  rate  is  40c.  on  $100  ? 

6.  If  a  hotel  is  insured  for  89000,  at  2  $,  what  is 
the  premium  ? 

7.  The  premium  on  a  factory,  insured  for  $7500,  is 
$75  ;  what  is  the  rate  ? 

8.  If  a  person  pays  8200  for  insuring  a  boat,  at 
2£$,  for  how  much  does  he  insure  her  ? 

9.  A  house  worth  $4000  is  insured  for  J  of  its  value, 
at  T3Q-  of  1  $.     If  it  burns  down,  how  much  will  the 
owner  receive  from  the  insurance  company,  and  how 
much  will  he  save  by  having  insured  ? 

10.  A  store  and  its  contents  are  insured  for  15  $  of 
their  value,  at  1  $>.     If  the  premium  is  $9,  and    the 
value  of  the  store  is  200  %  of  the  value  of  the  contents, 
how  much  is  the  store  worth  ?    What  is  the  value  of 
the  contents  ? 


TAXES.  125 

SECTION  77,— 1.  What  is  a  Tax?  Ans.  A  Tax  is 
a  sura  assessed  on  the  person,  property,  or  income  of 
an  individual,  for  the  support  of  government. 

2.  What  is  a  Poll-tax  ?     Ans.  A  tax  on  the  person, 
generally  a  uniform  sum  on  each  male  citizen  except 
such  as  are  exempted  by  law. 

3.  Ho\v  is  a  Property-tax  reckoned?     Ans.  At  a 
certain  </0  of  the  estimated  value  of  the  property. 

4.  What  is  Keal  Estate?    Ans.  Fixed  property, 
such  as  lands  and  houses. 

5.  What  is  Personal  Property  ?     Ans.  That  which 
is  movable  ;  cash,  notes,  furniture,  &c. 

6.  If  a  tax  of  $90  is  to  be  raised,  and  there  is  tax- 
able property  valued  at  $6000,  what  is  the  rate,  and 
what  must  A  pay,  who  has  property  valued  at  $1500  ? 

MODEL.  If  $6000  worth  of  property  has  to  pay  a  tax  of  $90,  $1 
must  pay  g^foo  of  $90,  or  6§So  of  $1,  or  ?fo.  yj5=H  hundredths, 
or  1£  per  cent. 

If  A  is  taxed  on  $1500  worth  of  property,  he  must  pay  1J-  %  of 
$1500,  or  $22.50.  Ans.  $22.50. 

7.  The  taxable  property  of  a  district  is  estimated 
at  $8000;  a  tax  of  $160  is  to  be  raised.     What  must 
A  pay  on  $425  personal  property  ?     What  must  B  pay 
on  $500  personal  property  and  $600  real  estate  ? 

8.  A  poll-tax  of  75c.  being  laid,  how  much  must  a 
person  pay  who  is  taxed  for  4  polls  ? 

9.  The  tax-rate  in  a  certain  town  being  -J-  of  1  $, 
what  will  be  the  tax-bill  of  a  person  who  has  real 
estate  valued  at  $1000  and  $600  -worth  of  personal 
property  ?      What  will  be  the  bill  of  one  who  has 
$2200  worth  of  personal  property  ? 


126  DUTIES. 

SECTION  78.— 1.  What  are  Duties?  Ans.  Duties 
are  taxes  on  imported  goods,  levied  for  tbe  support  of 
government. 

2.  How  are  duties  charged?     Ans.  Either  at  a 
certain  sum  on  each  yard,  gallon,  &c.,  of  the  article 
imported  ;  or  at  a  certain  per  cent,  on  the  cost  of  the 
article  in  the  country  where  it  was  bought. 

3.  What  is  the  duty  on  84500  worth  of  silks,  the 
rate  being  60  %  ? 

4.  Required  the  duty  on  5  hogsheads  of  molasses, 
containing,  after  leakage  is  deducted,  60  gallons  each, 
tbe  rate  being  8c.  a  gallon. 

5.  What  is  the  duty  on  10  hhd.  of  sugar,  weighing 
1200  Ib.  each,  12|#  being  allowed  for  the  weight  of 
the  casks,  and  the  duty  being  2c.  a  pound? 

6.  The  duty  on  sewing-silk  is  40  <£  on  the  cost. 
What  is  the  duty  on  100  Ib.  of  sewing-silk  bought  for 
$11  a  Ib.,  and  50  Ib.  bought  for  $12  a  Ib.  ? 

7.  How  much  must  sewing-silk  be  sold  for,  to  gain 
50  %  on  the  entire  cost,  if  it  was  bought  for  812  a  Ib., 
and  the  duty  on  it  was  40  <fa  ? 

8.  A  case  of  brandy,  containing  12  bottles,  was 
bought  for  824;  the  duty  was  8-3.60  a  gallon.     Five 
bottles  being  allowed  to  the  gallon,  what  must  be 
charged  per  bottle,  to  make  10  %  on  the  whole  cost  ? 

9.  How  many  pounds  of  raisins  does  a  merchant 
import,  if  the   duty  on   them,  at  2^  cents  a  pound, 
amounts  to  8115  ? 

10.  The  duty  on  8  packages  of  snuff,  averaging  10 
Ib.  net  weight  each,  is  840.     What  is  the  rate  of  duty 
per  pound  ? 


INTEREST.  127 


CHAPTER    TWELFTH. 

INTEREST. 

SECTION  79,— 1.  What  is  Interest?  Am.  Interest 
is  what  is  paid  for  the  use  of  money. 

2.  What  is  the  Principal?    Ans.  The  money  used, 
for  which  interest  is  paid. 

3.  What  is  the  Rate  ?    Ans.  The  per  cent,  paid  for 
the  use  of  the  principal  for  a  certain  time, — one  year, 
unless  some  other  time  is  specified. 

4.  What  is  the  Amount?    Ans.  The  sum  of  the 
principal  and  interest. 

A  person  borrows  $100  for  a  year,  and  pays  $6  for  its  use ;  the 
Principal  is  $100,  the  Interest  $6,  the  Rate  6  #,  the  Amount  $106. 

5.  At  7  fa  what  is  the  interest  of  $49,  for  1  yr.  ? 

MODEL.  7  #=TUO-  iffo  of  $49  is  49c.,  and  T£7  is  7  times  49c., 
or  343c.—  which  equals  $3.43.  Ans.  $3.43. 

6.  Find  the  interest  of  $900,  at  3  fa  for  1  yr. 

7.  Find  the  interest  of  $800,  at  6  fa  for  1  yr. 

8.  Find  the  interest  of  $350,  at  Q.fa  for  1  yr. 

9.  Find  the  interest  of  $700,  at  7  fa  for  1  yr. 

10.  Find  the  interest  of  $2000,  at  5  fa  for  1  yr. 

11.  Find  the  interest  of  $90.50,  at  4  fa  for  1  yr. 

12.  Find  the  interest  of  $50.50,  at  8  fa  for  1  yr. 

13.  Find  the  amount  of  $600,  for  1  yr.,  at  5£^. 

MODEL.  At  1  fa  the  interest  on  $600  would  be  $6,— at  5  fa  5 
times  $6,  or  $30,— and  at  £  fa  $•  of  $6,  or  $3.  $30 +  $3 =$33.  The 
interest,  at  6£  #,  is  $33;  and  the  amount  is  $600 +  $33,  or  $631. 
Ans.  $633. 


128  IXTEKEST. 

14.  Find  the  amount  of  $100,  for  1  yr.,  at 

15.  Find  the  amount  of  $480,  for  1  yr.,  at 
1G.  Find  the  amount  of  $820,  for  1  yr.,  at 

17.  Find  the  amount  of  $80,  for  1  yr.,  at  6  $. 

18.  Find  the  interest  of  8450,  for  1  yr.,  at  7  <f>. 

19.  Find  the  interest  of  $240,  for  1  yr.,  at  ^fc. 

20.  What  must  I  pay,  to  take  up  a  note  for  $100. 
with  interest  at  5  %  for  12  months  ? 

21.  A  person  lends  $1000  for  1  year,  at  7  <f>.     With 
another  $1000  he  buys  a  house,  which  at  the  end  of  a 
year  he  sells  for  $1100.     From  which  investment  does 
he  make  the  more  money,  and  how  much  more  ? 


SECTION  80.— 1.  What  is  the  interest  of  $200,  at 
7  £,  for  2  years  ? 

MODEL.     The  interest  on  $200,  at  7  #,  for  1  year,  is  $14 ;  and  for 
2  years  it  is  twice  $14,  or  $28.     Ans.  $28. 

2.  Find  the  interest  of  $120,  for  3  yr.,  at  5  fc. 

3.  Find  the  interest  of  $400,  for  6  yr.,  at  7  fc. 

4.  Find  the  amount  of  $700,  for  3  yr.,  at  6  $>. 

5.  Find  the  amount  of  $40,  for  5  yr.,  at  4£#. 

6.  Find  the  interest  of  $600,  for  4  yr.,  at  8  <f>. 

7.  Find  the  amount  of  $350,  for  2  yr.,  at  7$. 

8.  What  is  the  interest  of  $8000,  for  10  yr.,  at 
10  %  ?     Of  $300,  for  20  yr.,  at  5  %  ?     What  is  the  in- 
terest on  any  principal,  when  the  product  of  the  rate 
and  years  is  100  ? 

9.  Find  the  interest  of  $80,  for  3  mo.,  at  7  #. 
MODEL.     The  interest  of  $80,  at  7  fr  for  1  yr.,  is  $5.60;  and  for 

S  mo.,  which  is  £  of  a  year,  it  is  £  of  $5.60,  or  $1.40.     Ans.  $1.40. 


INTEREST.  129 

10.  At  8$,  what   is   the  interest  of  81000,  for  2 
months  ?     For  5  months  ? 

11.  At   7#,  what  is  the  interest  of  $1600,  for  3 
months  ?     For  9  months  ?     For  1  year  9  months  ? 

12.  At  5$,   what  is  the  amount  of  $300,  for  2 
'months  ?     For  1  yr.  2  mo.  ?     For  2  yr.  2  mo.  ? 

13.  At  4j-#,  what  is  the  amount  of  $200,  for  10 
months  ?     For  6  mo.  ?    For  3  yr.  6  mo.  ? 

14.  At  7  $,  what  is  the  interest  of  $60,  for  4  mo.  ? 
For  1  yr.  4  mo.  ?    For  11  mo.  ? 

15.  At  4#,  what  is  the  interest  of  $1000,  for  1  mo.  ? 
For  5  mo.  ?     For  8  mo.  ? 

16.  At  5  J-  #,  what  is  the  interest  of  $1200,  for  7  mo.  ? 
For  1  yr.  11  mo.  ? 

17.  At  7  #,  what  is  the  interest  of  $2400,  from  Oc- 
tober 3d  to  December  3d  of  the  same  year  ? 

18.  What  is  the  interest  of  $750,  from  February 
15th  to  August  15th  of  the  same  year,  at  6  %  ? 

19.  What  is  the  amount  of  $600,  from  November 
1st  to  June  1st  of  the  next  year,  at  5  %  ? 

20.  What  must  I  pay  for  the  use  of  $175,  from  May 
8th,  1866,  to  January  8th,  1868,  at  8  %  ? 

21.  February  1st  A  lent  B  $100,  and  on  March  1st 
$200  more.     Interest  being  allowed  at  6  $,  how  much 
should  B  pay  A  on  settling  up,  September  1st  of  the 
same  year  ? 

22.  A  person  depositing  $200  in  a  savings-bank, 
June  1st,  1867,  gets  5f0  interest.     Had  he  waited  till 
July  1st,  he  might  have  invested  it  differently  at  7  %  ; 
in  this  case,  how  much  better  off  would  he  have  been 
on  the  1st  of  June,  1868  ? 


130  INTEREST. 

SECTION  81. — 1.  What  is  the  interest  of  $600,  for 
10  days,  at  5  ^  '? 

NOTE.     In  calculating  interest,  30  days  are  allowed  to  the  month 
MODEL.     At  5  #,  the  interest  of  $600,  for  1  year,  is  830, — for  1 

month,  -fa  of  §30,  or  $2.50, — and  for  10  days,  which  is  £  of  1  month, 

^  of  $2.50,  or  83^c.     Arts.  SS^c. 

We  reject  fractions  of  a  cent  less  than  J,  and  call  £  or  more  an 

additional  cent ;  which  makes  the  above  answer  83c. 

2.  What  is  the  interest  of  $950,  at  6  fa  for  3  days  ? 
For  10  days  ? 

3.  What  is  the  amount  of  8800,  at  7  fa  for  12  days  ? 
For  15  days  ? 

4.  What  is  the  amount  of  $2000,  at  tyfa  for  8 
days  ?    For  20  days  ? 

5.  What  is  the  interest  of  $1200,  at  4  fa  for  5  days  ? 
At  7  fa  for  25  days  ? 

6.  What  is  the  interest  of  $900,  at  5  fa  for  4  days  ? 
At  6  fa  for  18  days? 

7.  What  is  the   amount   of  $450,  at  8  fa  for  27 
days  ?     At  7  #,  for  7  days  ? 

8.  What  is  the  interest  of  $7500,  at  5  fa  for  22 
days  ?     For  28  days  ? 

9.  What  is  the  interest  of  $85.50,  at  8  fa  for  15 
days?     For  18  days? 

10.  What  is  the  interest  of  $100,  at  7  fa  from  May 
1st,  1868,  to  May  22d,  1868  ? 

11.  What  is  the  amount  of  $250,  at  6£$,  from  Jan. 
1st,  1868,  to  Jan.  31st,  1868? 

12.  Borrowed,  May  28th,  $200,  and  June  3d,  $300. 
How  much  money  will  be  needed  to  pay  off  these 
debts,  with  interest  at  7  fa  June  12th  of  the  same  year? 


INTEREST.  131 

SECTION  82.— 1.  What  is  the  interest  of  $300,  at 
7  fa  for  1  yr.  6  mo.  6  da.  ? 

MODEL.  At  7  %,  the  interest  of  $300,  for  1  year,  is  $21.  For  1 
month,  the  interest  is  -^  of  $21,  or  $1.75, — for  6  months,  6  times 
$1.75,  or  $10.50, — and  for  6  days,  which  is  \  of  a  month,  \  of  $1.75, 
or  35c.  For  1  year  6  months  6  days,  therefore,  the  interest  is  $21  -f 
$10.50 +  35c.,  or  $31.85.  Ans.  $31.85. 

2.  What  is  the  interest  of  $900,  at  7  fa  for  1  year 
4  months  10  days  ? 

3.  What  is  the  amount  of  $480,  at  6  fa  for  9  mo.  5 
da.  ?    At  8  fa  for  the  same  time  ? 

4.  At  ±fa  what  is  the  amount  of  $1020  for  1  yr.  8 
mo.  9  da.  ?    For  2  yr.  8  mo.  9  da.  ? 

5.  At  5-J-^,  what  is  the  interest  of  $1000,  for  1  year 
3  days  ?     For  3  yr.  3  mo.  ? 

6.  At  5  fa  what  is  the  interest  of  $3200  for  2  yr.  3 
mo.  18  da.  ?     For  3  yr.  9  mo.  1  day  ? 

7.  At  1  %  a  month,  what  is  the  interest  of  $5000,  for 
21  days  ?     For  5  mo.  2  da.  ?     For  2  mo.  27  da.  ? 

8.  At  J  of  1  %  a  month,  what  is  the  interest  of 
$800  for  3  mo.  28  da.  ?     For  1  mo.  15  da.  ? 

9.  What  is  the  amount  of  $625,  at  10  fa  from  Oc- 
toher  1st,  1866,  to  April  4th,  1868? 

10.  A  person  bought  some  property,  Jan.  1st,  1868, 
for  $1000,  borrowing  the  money  to  pay  for  it,  at  6  fa 
For  how  much  must  he  sell  it  July  1st,  1868,  to  make 
10  f0  on  the  cost  and  interest  ? 

11.  A  house  was  bought  for  $8000, — $2000  cash, 
and  the  balance  to  be  paid  in  two  equal  semi-annual 
instalments,  with  interest  at  7  fa    What  was  the  whole 
amount  paid  ? 


132  INTEREST. 

SECTION  83.— 1.  What  is  meant  by  the  Legal  Rate 
of  interest  ?  Ans.  A  rate  fixed  by  law,  for  cases  in 
which  no  rate  is  specified. 

2.  In  what  States  is  the  legal  rate  7  %  ?     Ans.  In 
New  York,  New  Jersey,  Michigan,  Wisconsin,  Min- 
nesota, Kansas,  South  Carolina,  and  Georgia. 

3.  What  is  the  legal  rate  in  most  of  the  United 
States  ?    Ans.  In  most  of  the  United  States,  including 
all  the  New  England  States,  the  legal  rate  is  6$. 
Hence  it  is  important  to  know  the  shortest  methods 
of  computing  interest  at  6  fa 

4.  At  6  fa  what  is  the  interest  of  $1,  for  2  months  ? 
At  6  fa  the  interest  for  2  months  is  what  part  of  the 
principal  ? 

5.  Find  the  interest  of  $75,  at  6  fa  for  2  mo. 

MODEL.  At  6  #,  the  interest  for  2  months  is  TJo  of  the  princi- 
pal. 7-^0  of  $75  is  75c.  Ana.  75c. 

6.  Find  the  interest  of  $8300,  at  6  fa  for  2  mo. 

7.  Find  the  interest  of  $302.50,  for  2  mo.,  at  6  fa 

NOTE.  Dividing  a  sum  consisting  of  dollars  and  cents  by  100  is 
done  by  simply  moving  the  point  two  places  to  the  left.  TJ^  of 
8302.50  is  $3.025.  Ana.  $3.03. 

8.  Find  the  amount  of  $60.25,  for  2  mo.,  at  6  fa 

9.  Find  the  interest  of  $999,  for  2  mo.,  at  6  fa 

10.  At  6  fa  what  is  the  interest  of  $130.75,  for  2 
nonths  ?     The  amount  of  $8497,  for  60  days  ? 

11.  Find  the  interest  of  $2450,  for  16  mo.,  at  6  fa 

MODEL.  The  interest  for  2  months  is  -j-foy  of  the  principal,  or 
$24.50;  and  for  16  months  it  is  as  many  times  $24.50  as  2  mo.  is 
contained  times  in  16  mo.,  or  8.  8  times  $24.50  is  $196.  Ana.  $198. 


INTEEEST.  133 

12.  At   6$,  what  is  the.  interest  of  $88.90,  for  8 
months  ?     For  18  months  ?     For  1  year  6  months  ? 

13.  At  6$,  what  is  the  amount  of  $9000,  for  10 
months  ?     For  1  year  2  months  ? 

14.  At  6#,  what  is  the  interest  of  $4800,  for  11 
months  ?     For  7  months  ?     For  2  yr.  8  mo.  ? 

15.  At  6  $,  what  is  the  amount  of  $145,  for  20  mo.  ? 

16.  At  6  $,  what  was  the  interest  of  $700,  from 
July  7th,  1866,  to  October  7th,  1867  ? 

17.  How  much  was  to  pay  on  taking  up  two  notes, 
for  $200  each,  both  due  March  3d,  1868,  and  bearing 
interest  at  6  % ;  one  being  dated  Nov.  3d,  1867,  and 
the  other  Feb.  3d,  1867  ? 


SECTION  84.-— 1.  What  is  the  interest  of  $620,  for 
SO  days,  at  6  %  ? 

MODEL.  The  interest  for  60  days  is  jfa  of  the  principal,  or 
$6.20 ;  and  for  SO  days  (which  is  £  of  60  days)  it  is  £  of  $6.20,  or 
$3.10.  Ans.  $3.10. 

2.  Find  the  interest  of  $355,  for  30  days,  at  6  %. 

3.  Find  the  interest  of  $80,  for  30  days,  at  6  %. 

4.  Find  the  interest  of  $800,  for  3  days,  at  6  #. 

5.  What  is  the  interest  of  $650,  at  6  #,  for  30  days  ? 
For  3  days  ?     For  33  days  ? 

6.  What  is  the  interest  of  $2040,  at  6$,  for  90 
days  ?     For  3  days  ?     For  93  days  ? 

7.  What  is  the  amount  of  $18.60,  at  6$,  for  60 
days  ?     For  3  days  ?     For  63  days  ? 

8.  What  is  the  interest  of  $1100,  at  6  #,  for  33 
days  ?     For  63  days  ?     For  93  days  ? 


134  INTEREST. 

9.  What  is  the  interest  of  $1200,  for  1   year  10 
months  18  days,  at  6  %  ? 

MODEL.  The  interest  for  2  months  is  T^Q  of  the  principal,  or 
812,— for  1  yr.  10  mo.,  or  22  months,  it  is  11  times  $12,  or  $132,— 
and  for  18  days  (which  is  \%  of  2  months)  it  is  £J,  or  -,%,  of  812, 
which  is  $3.60.  Hence  for  1  yr.  10  mo.  18  da.  it  is  $132  + $3.80,  or 
$135.60.  Am.  $135.60. 

10.  At  6  foy  what  is  the  interest  of  $336,  for  2  years 
12  days  ?     For  1  yr.  7  mo.  10  da.  ? 

11.  At  6&  what  is  the  amount  of  $1500,  for  10 
months  15  days  ?    For  1  yr.  11  mo.  5  da.  ? 

12.  At  6#,  what  is  the  interest  of  §1800,  for  4 
months  5  days  ?     For  1  yr.  3  mo.  25  da.  ? 

1 3.  At  6  $,  what  is  the  interest   of  $2000,  for  8 
months  21  days  ?     For  3  yr.  4  mo.  8  da.  ? 

14.  At  6  fc,  what  is  the  interest  of  $2400  for  6  days  ? 
At  6  %,  the  interest  for  6  days  is  what  part  of  the 
principal  ? 

15.  At  6$,  what  is  the  interest   of  $8880   for  6 
days  ?     For  12  days  ?     For  3  days  ?    For  9  days  ? 

16.  Find  the  interest  of  $540,  for  2  mo.,  at  7  %. 

MODEL.  At  .6  %  it  would  be  $5.40 ;  at  1  #,  \  of  $5.40,  or  90c. ; 
and  at  7  #,  it  is  7  tunes  90c.,  or  $6.30.  Ans.  $6.30. 

Solve  this  Example  according  to  the  method  shown  in  Section 
80,  and  see  whether  the  answers  agree.  Solve  the  following  accord- 
ing to  both  methods. 

17.  Find  the  interest  of  $600,  for  10  mo.,  at  7  $. 

18.  Find  the  interest  of  $9000,  for  9  mo.,  at  5  <f>. 

19.  Find  the  amount  of  $420,  for  18  mo.,  at  4  <f>. 

20.  Find  the  interest  of  $6500,  for  14  mo.,  at  8  %. 

21.  Find  the  amount  of  $836,  for  8  mo.,  at  7#. 


INTEREST.  135 

SECTION  85.— 1.  What  part  of  a  year's  interest 
is  1  month's  interest  ? 

2.  Allowing  30  days  to  a  month,  what  part  of  a 
year's  interest  is  1  day's  interest  ? 

3.  How  many  days  are  there  in  a  year?     What 
part  of  1  year's  interest,  then,  would  1  day's  interest 
exactly  be  ? 

4.  By  taking  1  d"ay's  interest  as  -g-J-^  of  1  year's  in- 
terest (in  stead  of  -5^),  do  we  take  more  or  less  than 
the  exact  amount  ?    Is  the  difference  great  ? 

5.  Why,  do  you  suppose,  are  30  $ays  allowed  to 
the  month  in  computing  interest  ? 

6.  Required  the  interest,  by  the  shortest  method, 

Of  $70.50,  at  6  fa  for  60  days. 
Of  $1040,  at  6  fa  for  90  days. 
Of  $88.00,  at  8  fa  for  30  days. 
Of  $3750,  at  4  fa  for  63  days. 
Of  $225,  at  6  fa  for  2  yr.  6  mo. 
Of  $22.50,  at  5  fa  for  1  yr.  6  mo. 

7.  Required  the  amount,  by  the  shortest  method, 

Of  $3.47,      at  10  fa  for  10  years. 
Of  $1400,     at    8  fa  for  12  yr.  6  mo. 
Of  $219.90,  at    6  fa  for  16  yr.  8  mo. 
Of  $2000,     at    6  fa  for  1  yr.  1  mo.  1  da. 
Of  $860,       at    7  fa  for  1  yr.  8  mo.  9  da. 
Of  $900,       at    7  fa  for  7  mo.  7  da. 
Of  $400,       at    7  fa  for  3  mo.  10  da. 

8.  Jan.  1st,  A  sells  for  B  6  bales  of  cotton,  aver- 
aging 350  Ib.  each,  at  25c.  a  pound,     tfan.  21st,  he 
remits  to  B  $125.     How  much  is  due  to  B,  Feb.  21st, 
interest  being  allowed  at  6  $  from  the  day  of  sale  ? 


136  INTEREST. 

SECTION  86.—  1.  At  what  rate  will  8200  yield  89 
interest  in  1  year  ? 

MODEL.  At  1  £,  8200  will  yield  $2  interest  in  1  year  ;  if  it  yields 
£9,  the  rate  must  be  as  many  times  1  %  as  $2  is  contained  times  in 
$9,  or4£.  Ans.  4\%. 


2.  At  wThat  rate  must  $650  be  invested,  to  yield 
839  interest  annually  ?     To  yield  845.50  ? 

3.  At  what  rate  of  interest  will  83000  produce  8210 
a  year  ?     At  what  rate  will  it  produce  8120  ? 

4.  If  82500  produces  8200  interest   in  12  months, 
what  is  the  rat£  ? 

5.  A  person  lends  81000,  and  at  the  end  of  a  year 
receives  for  principal  and  interest  81065.    What  is  the 
interest  ?     What  rate  of  interest  does  he  receive  ? 

6.  If  a  person  receives  8481.50  for  a  loan  of  8450 
for  1  year,  what  rate  of  interest  does  he  get? 

7.  At  what  rate  must  £2000  be  invested,  to  amount 
to  £2060  in  lyear? 

8.  At  what  rate  will  §9000  yield  8675  interest  in  1 
year  6  months  ? 

MODEL.  1  yr.  6  mo.=l|  or  I  Jr-  If  tne  interest  for  f  years  is 
8675,  for  \  year  it  is  \  of  $675,  or  $225  ;  and  for  1  yr.  it  is  twice 
8225,  or  $450.  At  1  &  $9000  will  yield  $90  in  1  year  ;  if  it  yields 
$450,  the  rate  must  be  as  many  times  1  %  as  $90  is  contained  times 
in  $450,  or  5.  Am.  5  %. 

9.  At  what  %  will  8800  produce  860  in  1  yr.  3  mo.  ? 

10.  At  what  %  will  8900  give  8126  interest  in  2  yr.  ? 
11..  At  what  rate  will  8240  produce  822.40  interest 

in  1  year  4  iftonths  ? 

12.  At  what  rate  will  8606  produce  $60.60  interest 
in  60  days  ? 


INTEREST.  137 

13.  At  what  rate  must  $1000  be  invested,  to  yield 
$50  interest  in  10  mo.  ?     To  yield  $200  in  4  yr.  ? 

14.  At  what   rate  is  $1100  invested,  if  it  yields 
$38.50  interest  semi-annually  ? 

15.  At  what  rate  must  $2500  be  invested,  to  amount 
to  $2850  in  2  years  ? 

16.  If  the  amount  of  $6700,  for  1   yr.   8  mo.,  is 
$7370,  what  is  the  rate? 

17.  At  what  rate  must  $490  be  put  at  interest,  to 
amount  to  $627.20  in  3  yr.  6  mo.  ? 

18.  At  what  rate  will  the  interest  on  any  sum  equal 
the  principal  in  10  years  ? 

MODEL.     The  interest  will  equal  the  principal  in  1  year  at  100  <£, 
and  in  10  years  at  -^  of  100  #,  or  10  %.    Am.  10  %. 

19.  At  what  rate  will  $100  produce  $100  interest  in 
4  years  ?     In  6  years  ?     In  20  years  ? 

20.  At  what  rate  will  the  interest  on  a  certain  sum 
equal  the  principal  in  12  years  ?     In  5  years  ? 

21.  At  what  rate  will  any  principal  double  itself  in 
2  years  ?    In  8  years  ?    In  15  years  ? 


SECTION  87.— 1.  How  long  will  it  take  $200,  at 

5  <fc,  to  produce  $22  interest  ? 

MODEL.  In  1  year,  at  5$,  $200  will  produce  $10,  and  to  produce 
$22  will  take  it  as  many  years  as  $10  is  contained  times  in  $22,  or 
2£.  2J-  years =2  yr.  2  mo.  12  da.  Ans.  2  yr.  2  mo.  12  da. 

2.  How  long  will  it  take  $650,  at  6  <fc,  to  yield  $78 
interest  ?     How  long,  to  yield  $6.50  ? 

3.  How  long  will  it  take  $100,  at  7$,  to  yield 
$17.50  interest  ?     How  long,  to  yield  $3.50  ? 


138  IXTEREST. 

4.  How  long  will  it  take  §3000,  at  8  £,  to  produce 
$400  interest  ?     To  produce  §260  ? 

5.  How  long  will  it  take  $200,  at  1  %  a  month,  to 
produce  $1  ?    To  produce  $5  ? 

6.  At  £$  a  month,  in  what  time  will  §500  produce 
§15? 

7.  How  long  must  §1600  be  at  interest,  at  7$,  to 
yield  §448  interest  ?     To  amount  to  $2048  ? 

8.  In   what  time  will  §1250,  at   4$,  amount   to 
$1312.50?     To  §1500? 

9.  At  7  $,  in  what  time  will  the  interest  on  any 
sum  equal  the  principal  ? 

MODEL.  The  interest,  being  7  %  of  the  principal  in  1  year,  will 
be  100  %  of  it,  or  equal  to  it,  in  as  many  years  as  7  is  contained 
times  in  100,  or  14  f.  Ans.  142-  years. 

10.  At  8  #,  how  long  will  it  take  §50  to  yield  §50 
interest  ?    How  long,  at  4  %  ?     How  long,  at  5  $  ? 

11.  In  what  time  will  any  principal  double  itself  at 
6  %  ?     At  10  %  ?     At  1  fc  a  month  ? 

12.  In  what  time  will  any  principal  gain  50^  of 
itself  at  7  %  ?    At  5  %  ?     At  4£  %  ? 

13.  At  6  $,  how  long  will  it  take  any  principal  to 
gain  \  of  itself?     To  gain  }  of  itself? 


SECTION  88. — 1.  What  sum  put  at  interest  for  2 
yr.  6  mo.,  at  6  $,  will  produce  §225  ? 

MODEL.  The  interest  on  $1,  for  1  year,  at  6  £,  is  6c. ;  and  for  2 
yr.  and  6  mo.,  2J-  times  6c.,  or  15c.  The  required  principal  is  there- 
fore as  many  times  $1  as  15c.  (which  is  -jVnr,  or  -3 
tained  times  in  $225,  or  1500  times.  Ans.  $1500. 


INTEREST.  139 

2.  What  principal,  in  9  months,  at  8  $,  will  yield 
$72  interest? 

3.  What  sum  must  be  invested,  at  7  $,  to  yield 
$115.50  annually? 

4.  A  lady  wishes  to  provide  a  semi-annual  income 
of  $200  for  her  son  ;  how  much  must  she  invest  in  his 
name,  at  5  %  ? 

5.  What  principal  invested  at  6  %  will  yield  $8.25 
interest,  in  1  year  10  months  ? 

6.  What  sum  put  at  interest  at  6-J -%  will  produce 
$52  in  4  years  ? 

7.  What  principal,  at  1  %  a  month,  will  in  3  months 
15  days  yield  $17.50  interest  ? 

8.  What  principal,  at  £  %  a  month,  will  in  4  months 
amount  to  $408  ? 

NOTE.  At  \.  %  a  month,  in  4  months  T£<j  or  •£$  of  the  principal 
equals  the  interest.  The  principal  being  f  ft  of  itself,  the  amount 
must  be  fft+s1^,  or  Ji,  of  the  principal.  The  question  therefore  be- 
comes, $408  is  % -J-  of  how  much  ? 

9.  What  principal,  at  7  #,  will  amount  to  $96.80  in 
3  years  ? 

10.  A  person,  having  borrowed  some  money  for  1 
yr.  3  mo.,  repays  it  with  interest  at  8  $,  the  amount 
being  $352.     What  was  the  sum  borrowed  ? 

11.  What  principal  will  double  itself  in  10  years, 
at  10  #? 

12.  A  person,  having  inherited  25  <f0  of  his  father's 
property,  put  it  out  at  interest  at  6  %.  With  the  amount, 
at  the  end  of  2  years,  he  was  able  to  buy  \  of  a  mill 
valued  at  $4480.     How  much  property  was  left  by  his 
father? 


140  DISCOUNT. 


CHAPTER     THIRTEENTH. 

DISCOUNT. 

SECTION  89.— 1.  What  is  Discount?  Am.  An 
allowance  made  for  the  payment  of  money  before  it  is 
due. 

2.  At  2^  $,  what  will  be  the  discount  for  cash  on  a 
bill  of  $800  ? 

3.  A  person  buys  a  bill  of  goods  amounting  to 
$1500,  and  can  have  4  months'  credit  or  a  discount  of 
5  %  for  cash.*    If  he  chooses  the  latter,  what  will  be 
the  discount,  and  how  much  cash  will  pay  his  bill  ? 

4.  A  merchant  buys  $2400   worth  of  goods,  3$ 
being  allowed  for  cash.     How  much  cash  will  pay  his 
bill,  and  what  will  the  discount  amount  to  ? 

5.  Bought  50  pieces  of  cassimere,  averaging  36 
yards  each,  at  $2  a  yard,  5  %  off  for  cash.     For  how 
much  must  a  check  be  drawn,  to  pay  the  bill  ? 

6.  A  person,  having  bought  a  bill  of  hardware,  ob- 
tained a  deduction  of  $22.50  by  paying  cash  at  a  dis- 
count of  2£  $  ;  what  was  the  amount  of  his  bill  ? 

7.  If  by  paying  cash  a  merchant  gets  a  discount  of 
890  on  a  bill  of  $3000,  what  is  the  rate  of  discount  ? 

8.  A  publisher  sells  a  ten-dollar  annual  for  30  %  less 
than  its  retail  price,  and  then  makes  a  discount  of  5  % 
for  cash.     What  must  the  purchaser  sell  it  for,  in  order 
to  make  20  %  ? 

*  This  does  not  mean  at  the  rate  of  5  %  a  year,  but  5  %  on  the  face  of  tL* 
bill,  without  reference  to  time. 


TRUE   DISCOUNT.  141 

SECTION  90.— 1.  How  does  True  Discount  differ 
from  the  discount  treated  of  in  the  last  Section  ?  Ans. 
In  computing  True  Discount,  time  is  taken  into  ac- 
count. 

2.  What  is  the  Present  Worth  of  a  sum  due  at  a 
future  time  without  interest  ?     Ans.  Such  a  sum  as 
put  at  interest  for  the  given  time  will  amount  to  the 
debt. 

3.  What  is  the  True  Discount?    Ans.  The  differ- 
ence between  the  Present  Worth  and  the  debt. 

4.  If  I  owe  $53,  due  in  12  months  without  interest, 
what  sum  now  paid  would  discharge  the  debt,  money 
being  worth  6  %,  and  what  is  the  true  discount  ? 

MODEL.  In  12  months,  at  6  $,  $1  would  amount  to  $1.06, — that 
is,  to  |JJ,  or  f  J,  of  itself.  $53  is  therefore  f  §  of  the  present  worth  ; 
-,f0-  is  -Q^J  of  $53,  or  $1  ;  and  £{y,  or  the  present  worth,  is  50  times  $1, 
or  $50.  The  true  discount  is  $53— $50,  or  $3.  Ans.  Present  worth, 
$50 ;  true  discount,  $3. 

5.  What  is  the  present  worth  of  $228,  due  2  years 
hence,   without    interest,   when   money  brings   7  %  ? 
What  is  the  true  discount  ? 

6.  At  5  f0,  what  is  the  present  worth  of  $90,  due  in 

4  years  ?     What  is  the  true  discount  ? 

7.  At  6  $,  what  is  the  present  worth  of  $303,  due 
in  60  days  ?     What  is  the  true  discount  ? 

8.  At  8  #,  what  is  the  present  worth  of  $1120,  due 

5  years  hence  ? 

9.  A  merchant,  having  bought  40  clocks,  at  $25.50 
apiece,  on  4  months'  credit,  afterwards  gets  a  discount 
of  3  %  for  cash.     By  how  much  does  this  discount  ex- 
ceed the  true  discount,  at  6  %  ? 


142  BAXK   DISCOUNT. 

SECTION  91.— 1.  What  is  Bank  Discount?  Ans. 
An  allowance  made  to  a  bank  for  cashing  a  note  be- 
fore it  is  due. 

2.  How  is  Bank  Discount  computed  ?    Ans.  At  a 
certain  %  on  the  face  of  the  note,  which  the  bank  re- 
tains, paying  over  the  balance  to  the  owner.     The 
balance  thus  paid  over  is  called  the  Proceeds. 

3.  A  bank  discounts  a  note  for  8200,  which  will 
mature  in  3  mo.,  at  7  % ;  what  is  the  discount,  and 
what  are  the  proceeds  ? 

MODEL.  At  7  £,  the  bank  discount  on  $200,  for  1  year,  would  be 
$14,  and  for  3  months  it  is  \  of  $14,  or  $3.50.  The  proceeds  are 
$200—$3.50,  or  $196.50.  Am.  Discount,  $3.50  ;  proceeds,  $196.50. 

4.  At  6  $,  what  is  the  bank  discount  on  a  note  for 
$1000,  to  mature  in  63  days?     On  a  note  for  $1200, 
to  mature  in  33  days? 

5.  Required  the  proceeds  of  a  note  for  $700,  ma- 
turing Sept.  7th,  and  discounted  the  7th  of  the  previ- 
ous June,  at  6  <fo. 

6.  A  person  invested  the  proceeds  of  a  note  for 
$1200,  discounted  for  4  months,  at  6  #,  in  flour  at  $8 
a  barrel.     How  many  barrels  did  he  buy  ? 

7.  What  is  the  bank  discount  on  a  note  for  $900, 
to  run  30  days,  at  7  %  ? 

8.  The  bank  discount,  for  2  months,  on  a  $300 
note,  was  $3.50 ;  what  was  the  rate  ? 

9.  The  proceeds  of  a  $600  note,  discounted  for  90 
days,  were  $591  ;  what  was  the  rate  ? 

10.  The  proceeds  of  a  note  for  $1000,  discounted 
at  6  #,  were  sufficient  to  pay  for  98  acres  of  land,  at 
$10  an  acre.     How  long  had  the  note  to  run  ? 


STOCKS.  143 

CHAPTER     FOURTEENTH. 

STOCKS, -U.    S.     SECURITIES. 

SECTION  92.— 1.  What  is  a  Bond?  Ans.  A  Bond 
is  a  written  instrument  by  which  one  party  binds  him- 
self to  pay  another  a  certain  sum. 

2.  What  is  meant  by  Stocks?     Ans.  Stocks  is  a 
general  term  applied  to  Government  or  State  bonds, 
and  the  capital  of  incorporated  companies. 

3.  How  is  stock  divided  ?     Ans.  Into  shares,  gen- 
erally of  $100  each. 

4.  When  is  a  stock  at  par?    Ans.  When  it  sells 
for  its  nominal  value.     If  it  sells  for  more  than  its 
nominal  value,  it  is  above  par,  or  at  a  premium ;  and 
if  less,  below  par,  or  at  a  discount. 

5.  What  are    100  shares  of  Erie  Railroad  stock 
worth,  at  69J  ? 

MODEL.     At  69£,  one  share  ($100)  is  worth  $69.50;  and  100 
shares  are  worth  100  times  $69.50,  or  $6950.     Ans.  $6950. 

NOTE.     Take  $100  for  a  share,  unless  it  is  otherwise  specified. 

6.  What  are  100  shares  of  Chicago  and  North- 
western worth,  at  67  ? 

7.  What  is  the  value  of  50  shares  of  N".  Y.  Central, 
at  125  (25  %  above  par)  ? 

8.  How  much  are  100  shares  of  Pacific  Mail  worth, 
at  8  f0  below  par  ? 

9.  If  N.  J.  Central  is  selling  at  a  premium  of  16 £  %, 
how  much  are  80  shares  worth  ? 

10.  If  100  shares  of  Milwaukee  and  St.  Paul  are 
worth  $6500,  how  much  below  par  is  the  stock  ? 


144  STOCKS. 

SECTION  93.— 1.  What  is  a  Stock-broker?  Ans. 
One  who  buys  and  sells  stock  for  others,  at  a  charge 
usually  of  £  of  1  $  on  the  par  value  of  the  stock  bought 
or  sold. 

2.  What  is  the  brokerage  on  100  shares  of  Hudson 
River  R.  R.  stock,  bought  at  137  ? 

MODEL.     %%  on  1  share  of  $100  is  J  of  a  dollar,  and  on  100 
shares  it  is  100  times  $£,  or  $25.     Ans.  $25. 

XOTE.     Reckon  brokerage  at  £  #,  unless  it  is  otherwise  specified. 

3.  What  is  the  brokerage  on  300  shares  of  Cleve- 
land and  Toledo,  sold  at  106  ? 

4.  What  is  the  brokerage  for  buying  and  selling 
150  shares  of  Great  Western? 

5.  For  what  %  advance  must  a  person  sell  stock,  to 
cover  the  brokerage  for  buying  and  selling,  and  make 
Ijf? 

6.  What  is  the  cost,  including  brokerage,  of  100 
shares  of  Western  Union  Telegraph  stock,  at  40  ? 

7.  How  much  cash  should  be  given  with  100  fifty- 
dollar  shares  of  a  mining  company,  selling  at  25,  for 
50  shares  of  Ohio  and  Mississippi,  worth  30  ? 

8.  What  is  the  profit,  over  and  above  brokerage,  on 
100  shares  of  stock  bought  at  94,  and  sold  at  97  ? 

9.  What  is  the  loss,  including  brokerage,  on  50 
shares  of  stock  bought  for  67,  and  sold  for  60  ? 

10.  A  person  has  bought  a  hundred  shares  of  stock 
at  99  ;  to  make  $500  on  his  purchase,  what  must  he 
sell  it  for  ? 

11.  A  large  dealer  in  stocks  arranges  with  a. broker 
to  buy  and  sell  for  him  at  -J-  %.     What  will  the  broker- 
age be  on  300  fifty-dollar  shares  ? 


STOCKS.  145 

12.  Fifty  shares  of  bank  stock  were  bought  at  par, 
i*nd  sold  at  103.     What  was   the  profit,  brokerage 
being  paid  on  each  transaction  ? 

13.  A  person  realized  $200  (leaving  brokerage  out 
of  account)  by  selling  100  shares  of  stock  at  75.     At 
what  rate  did  he  buy  them  ? 

14.  By  selling  200  shares  of  stock  at  81  £,  a  person 
lost  $450.     What  did  they  cost  him  ? 

15.  How  many  shares  20  %  below  par  can  be  bought 
for  $6000,  leaving  brokerage  out  of  account  ? 


SECTION  94,— 1.  What  is  a  Dividend?  Arts.  A 
sum  paid  from  the  earnings  of  a  company  to  those 
who  hold  its  stock. 

2.  How  is  a  dividend  reckoned  ?    Am.  At  a  cer- 
tain <fc  on  the  par  value  of  the  stock. 

3.  A  bank  declares  a  dividend  of  7  %.    What  will 
a  person  receive  who  holds  stock  worth  at  par  $8000  ? 

4.  A  railroad  company  having  declared  a  dividend 
of  3  $,  how  much  will  a  party  who  owns  50  shares  re- 
ceive ? 

5.  A  ferry  company  pays  a  dividend  of  1  %  a  month. 
How  much  will  a  person  who  owns  300  twenty-five- 
dollar  shares  receive  in  a  year  ? 

6.  A  person  who  bought  some  stock  at  50,  receives 
from  it  a  yearly  dividend  of  6  <fc.     What  per  cent,  does 
he  get  on  his  investment  ? 

MODEL.    Each  $100  share  draws  a  dividend  of  $6,  and,  being 
bought  at  50,  cost  $50.    He  therefore  makes  -6fio,  or  -fj^-.    Ans.  12%. 
13 


146  STOCKS. 

7.  If  a  person  buys  some  stock  at  96,  and  receives 
from  it  yearly  dividends  of  6  $,  what  per  cent,  does  he 
get  on  the  investment  ? 

8.  What  per  cent,  on  the  investment  will  be  real- 
ized from  stock  bought  at  60,  and  paying  a  semi-annual 
dividend  of  2£<£? 

9.  If  50  shares  of  stock  draw  a  dividend  of  $400, 
what  is  the  rate  of  dividend  ?     If  said  stock  was 
bought  at  160,  what  fc  does  it  pay  on  the  investment? 

10.  At  what  rate  must  bonds  that  pay  6  %  annually 
be  bought,  to  yield  8  $  on  the  investment  ? 

MODEL.  Each  $100  of  bonds  pays  $6,  which  must  be  8  %  of  the 
required  rate.  If  $6  is  T§Q,  or  gg,  of  the  required  rate,  gV  is  |  of 
•  $6,  or  $3  ;  and  fjj-  is  25  times  $3,  or  $75.  Ans.  75. 

11.  At  what  rate  must  5  %  bonds  be  purchased,  to 
pay  8  %  on  the  investment  ? 

12.  A  person  receiving  $300  dividend  on  100  shares 
of  stock,  gets  6  $  on  his  investment.     How  much  be- 
low par  did  he  buy  the  stock  ? 


SECTION  95.— 1.  When  gold  is  at  140,  how  much 
in  greenbacks  will  it  take  to  buy  $250  in  gold  ? 

MODEL.  At  140,  $100  in  gold  will  cost  $140  in  greenbacks ;  and 
$250  in  gold  will  cost  2£  times  $140,  or  $350.  Am.  $350. 

NOTE.  Since  1861,  gold  and  silver  have  commanded  a  premium ; 
that  is,  $1  in  gold  or  silver  has  been  worth  more  than  $1  in  currency. 

2.  A  merchant,  having  to  pay  his  duties  in  gold, 
needs  for  this  purpose  $300.  What  will  it  cost  in  cur- 
;rency,  when  gold  is  at  114  ? 


GOLD   AND    CURRENCY.  147 

3.  Having  $450  in  gold,  a  person  sold  it  at  136|-. 
How  much  more  would  he  have  got  for  it,  had  he 
waited  till  the  next  week,  when  gold  rose  to  141|? 

4.  What  must  be  paid  for  10  gold  eagles,  when 
gold  is  at  a  premium  of  12J  ? 

5.  Bought  10  Swiss  watches,  at  $100  each  in  gold. 
What  sum  in  currency  will  pay  the  bill,  gold  being  138  ? 

6.  A  person,  having  bought  some  jewelry,  has  the 
choice  of  paying  $6000  in  gold,  or  $8420  in  current 
funds.     Gold  being  140,  which  had  he  better  do  ? 

7.  When  gold  is  150,  how  much  gold  can  be  bought 
for  $600  in  currency  ? 

8  Having  $4200  in  currency,  a  person  invests  it  in 
gold  at  110,  which  he  sells  the  next  day  at  111 ;  how 
much  does  he  make  ? 

9.  A  merchant  changes  $650  currency  into  gold,  to 
pay  the  duty  on   some  drygoods.     If  gold  is  worth 
130,  how  much  does  he  buy? 

10.  A  person  who  had  received  a  3$  dividend  on 
90  shares  of  railroad  stock,  bought  half  eagles  with  it. 
How  many  did  he  buy,  if  gold  was  135  ? 

11.  How  much  more  gold  can  be  bought  for  $6000, 
when  gold  is  120,  than  when  it  is  150  ? 

12.  When  gold  stood  at  180,  how  much  currency 
was  $900  in  gold  worth  ?     How  much  gold  was  $900 
in  currency  worth  ? 

13.  With  gold  at  160,  how  much  more,  in  gold,  is 
a  person  worth  who  has  30  double-eagles,  than  one 
who  has  $800  in  greenbacks  ? 

14.  Twenty  eagles  were  bought  for  $280  in  cur- 
rency ;  how  did  gold  stand  ? 


148  U.  S.  SECURITIES. 

SECTION  96. — 1.  What  is  meant  by  Government 
Bonds  or  U.  S.  Securities?  Ans.  Bonds  issued  by  the 
United  States  Government. 

2.  What  are  Five-twenties?     Ans.  U.  S.  Bonds, 
bearing  interest  at  6  %  in  gold,  payable  in  not  less 
than  five  or  more  than  twenty  years  from  their  date. 

3.  What  are  Ten-forties  ?    Ans.  IT.  S.  Bonds,  bear- 
ing interest  at  5  %  in  gold,  payable  in  not  less  than 
ten  or  more  than  forty  years  from  their  date. 

4.  What   are  U.  S.  Currency  6's  ?    Ans.   U.  S. 
Bonds  payable  in  1895  and  thereafter,  bearing  inter- 
est at  6$  in  currency. 

5.  What  cost  $1000  of  five-twenties,  at  106  ? 

6.  At  103 J,  what  are  $5000  of  10-40's  worth? 

7.  A  person  who  had  100  shares  of  Erie  Railroad 
stock,  sold  them  at  70,  paying  brokerage,  and  bought 
10-40's  to  the  amount  of  $6000,  at  103  ;  how  much  of 
the  proceeds  of  his  stock  had  he  left  ? 

8.  A  person  who  had  $2200  in  currency,  bought 
two  5-20  bonds  of  $1000  each,  at  108.     How  much 
currency  had  he  left  ? 

9.  If  I  get  a  note  for  $2500  discounted  at  a  bank, 
for  60  days,  at  6  $,  how  much  more  than  the  proceeds 
of  this  note  will  I  need,  to  buy  $2500  in  U.  S.  Cur- 
rency 6's,  at  115  ? 

10.  What  amount  of  5-20's,  at  110,  can  be  bought 
for  $3300  ? 

11.  What  amount  of  10-40's,  at  104,  will  $5200  buy? 

12.  A  person  sold  $5000  in  gold,  at  115,  and  with 
the  proceeds  bought  six  5-20  bonds  of  $1000  each,  at 
110.     How  much  had  he  left  ? 


U.    S.    SECURITIES.  149 

SECTION  97. — 1.  What  income  in  gold  will  a  per- 
son annually  receive  from  $2000  of  five-twenties? 
From  $8000  of  ten-forties  ? 

2.  What  income  in  gold  will  a  person  receive  half- 
yearly  from  $3500  of  ten-forties  ?     From  $7500  of  five- 
twenties  ? 

3.  When  gold  is  at  138,  what  is  the  semi-annual 
income  in  currency  from  $5000  of  five-twenties  ? 

MODEL.  The  income  for  1  year  is  Tgn,  an^  f°r  half  a  year  Tjf(f, 
of  $5000,— which  is  $150,  in  gold.  If  gold  is  at  138,  $100  in  gold 
is  worth  $138  in  currency;  and  $150  in  gold  is  worth  1£  times  $138 
in  currency,  or  $207.  Ans.  $207. 

4.  When  gold  is  115,  what  is  the  value  in  currency 
of  one  year's  interest  on  two  five-twenty  bonds  of 
$1000  each  ? 

5.  When  gold  is  140,  what  is  the  value  in  currency 
of  six  months'  interest  on  six  ten-forty  bonds  of  $500 
each? 

6.  What  is  the  yearly  income  from  $1000  of  6's  ? 

7.  What  amount  of  five-twenties  will  yield  a  yearly 
income  of  $540  in  gold  ? 

8.  How  much  must  be  invested  in  ten-forties,  to 
yield  $100  in  gold  every  six  months  ? 

9.  A  person  had  equal  amounts  invested  in  five- 
twenties  and  ten-forties.      If  his  income  from  these 
bonds  was  $330  a  year  in  gold,  what  amount  of  each 
kind  had  he  ? 

10.  B  has  equal  amounts  invested  in  5-20's,  10-40's,, 
and  U.  S.  Currency  6's.     If  his  5-20's  yield  $60  a  year 
in  gold,  what  will  be  his  whole  income  from  these 
bonds  in  currency,  when  gold  is  112  ? 


150  MISCELLANEOUS   EXAMPLES. 


CHAPTER     FIFTEENTH. 

MISCELLANEOUS      EXAMPLES. 

SECTION  98. — 1.  What  is  the  cost  of  a  draft  on 
Charleston  for  $2250,  at  1  %  premium  ? 

2.  What  is  the  cost  of  a  draft  on  Cincinnati  for 
$1000,  at  \<f>  discount  ? 

3.  If  a  franc  is  worth  19^c.,  what  is  the  value  of 
1000  francs  in  U.  S.  currency? 

4.  The  pound  sterling  of  Great  Britain  is  worth 
$4.86-^,  and  the  Canada  dollar  is  worth  $1,  in  U.  S. 
gold.     How  many  Canada  dollars  are  equivalent  to 
500  pounds  sterling? 

5.  How  many  Canada  dollars  are  100  eagles  worth  ? 

6.  A  grocer  mixes  100  Ib.  of  coffee,  at  24c.,  with 
20  Ib.,  at  15c. ;  what  is  the  mixture  worth  per  Ib.  ? 

MODEL.  100  Ib.,  at  24c.  a  pound,  are  worth  $24.  20  Ib.,  at  15c., 
are  worth  $3.  The  whole,  120  Ib.,  is  therefore  worth  $24  +  $3,  or 
$27 ;  and  1  Ib.  is  worth  fa  of  $27,  or  22|c.  Ans.  22^c. 

7.  If  10  gall,  of  brandy,  at  $4  a  gallon,  were  mixed 
with  5  gall.,  at  $10  a  gallon,  how  much  a  gallon  was 
the  mixture  worth  ? 

8.  If  a  train  of  cars  goes  20  miles  an  hour  for  2 
hours,  and  then  30  miles  an  hour  for  3  hours,  what  is 
its  average  rate  for  the  whole  time  ? 

o 

9.  If  a  boat  runs  30  miles  in  2  h.,  and  then  15  miles 
in  1^  h.,  what  is  its  average  speed  ? 

10.  A  grocer  mixes  20  Ib.  of  tea  that  cost  70c.  a  Ib. 
with  20  Ib.  that  cost  $1.10  a  Ib.     At  what  price  per  Ib. 
must  he  sell  the  mixture,  to  make  20$  ? 


MISCELLANEOUS    EXAMPLES.  151 

11.  Two  thousand  bushels  of  wheat,  worth  $2.50  a 
bushel,  were  mixed  with  1000  bu.,  worth  $1.90.    How 
much  a  bushel  was  the  mixture  worth  ? 

12.  An  equal  number  of  geese,  chickens,  and  tur- 
keys, were  sold  for  $15.     Each  goose  brought  90c., 
each  chicken  50c.,  and  each  turkey  -^  of  the  cost  of  a 
goose  and  chicken  ;  how  many  of  each  were  sold  ? 

13.  A  person  bought  some  property  for  $2000  on 
the  10th  of  Jan.,  and  on  the  10th  of  May  following 
sold  it  for  $2200.     What  </0  did  he  make,  and  how 
much  better  off  was  he  than  if  he  had  loaned  his 
money  at  7  ^  for  the  same  time  ? 

14.  4  times  f  of  36  is  £  of  what  number  ? 


SECTION  99.-— 1.  How  much  will  a  pile  of  wood, 
16  feet  long,  4  feet  wide,  and  8  feet  high,  cost,  at  $6  a 
cord? 

2.  If  A  can  do  a  certain  piece  of  work  in  f  of  an 
nour,  and  B  can  do  it  in  $  of  an  hour,  how  long  will 
it  take  both  to  complete  the  job  after  B  has  been 
working  20  minutes  ? 

3.  The  sum  of  two  numbers  which  are  to  each 
other  as  5  to  4,  is  36  ;  what  are  the  numbers  ? 

4.  The  difference  between  two  numbers  which  are 
to  each  other  as  5  to  9,  is  24 ;  what  are  the  numbers  ? 

5.  A  rockaway  was  sold  for  $220,  at  a  profit  of 
10  %  ;  at  what  price  would  it  have  brought  a  profit  of 

6.  What  principal,  at  4  %,  for  2  years,  will  yield  as 
much  interest  as  $200,  at  6  <g,  for  3  years  ? 


152  MISCELLANEOUS    EXAMPLES. 

7.  A  commission-merchant  sold  a  consignment  of 
goods  for  $6000.     If  he  paid  $450  expenses,  and  his 
commission  was  2  J  %  on  the  sales,  how  much  should 
he  remit  to  the  consignor  ? 

8.  A  person  sold  two  houses  for  $3960  each,  making 
10$  on  one,  and  losing  10$  on  the  other.     Taking 
both  sales  into  account,  what  was  his  gain  or  loss  ? 

9.  What  principal  will,  in  2  years,  at  7  $,  amount 
to  a  sum  sufficient  to  buy  76  acres  of  land,  at  $30  an 
acre  ? 

10.  The  owner  of  a  farm  let  it  out  to  a  party  to 
work  on  shares,  allowing  him  40  %  of  all  he  raised. 
How  many  bushels   of  potatoes   were   raised,  if  the 
owner's  share  was  480  bushels  ? 

11.  A  person  who  received  75$  of  the  rent  of  a 
hotel,  with  50  %  of  his  income  for  1  year  from  this 
source  bought  \  of  a  mill.     If  the  hotel  rented  for 
$4000  a  year,  how  much  was  the  mill  worth  ? 

12.  Four-fifths  of  a  vessel  was  sold  for  $5760,  at  a 
loss  of  10  <fc.     How  much  would  the  whole  vessel  have 
had  to  sell  for,  to  bring  a  profit  of  10  %  ? 

13.  A  and  B  had  different  amounts  of  five-twenties, 
A's  being  to  B's  as  3  to  5.     "When  gold  was  at  150, 
the  two  realized  $720  yearly  interest  in  currency  from 
these  bonds  ;  what  amount  had  each  ? 

14.  A  man  whose  money  was  invested  at  6  $,  suc- 
ceeded in  changing  the  investment  so  as  to  get  7  $, 
and  found  that  it  made  a  difference  of  $37.50  in  his 
gemi-annual  income.     How  much  money  had  he? 

15.  At  how  much  a  pound  must  guano  be  sold,  to 
make  20  $,  if  it  cost  $85  a  ton  ? 


MISCELLANEOUS    EXAMPLES.  153 

SECTION  100, — 1.  A's  money  was  invested  at  5  #, 
B's  at  6  $,  and  C's  at  7  %.  The  sums  invested  were  to 
each  other  as  1,  2,  and  3.  If  their  yearly  incomes 
from  these  investments  were  together  $380,  how  much 
had  each  invested? 

2.  A  certain  principal,  at  5  $,  amounted  in  a  year 
to  $8.75  less  than  it  would  have  amounted  to  in  a  year 
and  a  half.     What  was  the  principal  ? 

3.  Three-fourths  of  John's  age  is  1  year  less  than 
Henry's,  and  1  year  more  than  Daniel's.     The  sum  of 
their  ages  being  50  years,  how  old  is  each  ? 

4.  A  person  left  some  money  to  be  divided  between 
three  of  his  relatives  in  the  proportion  of  £,  ^,  and  \. 
The  money  was  invested  at  6  $,  and  brought  $234 
yearly  interest.     How  much  of  the  principal  should 
each  receive  ? 

5.  D  sold  a  horse  to  E  at  a  profit  of  50  % ;  E  sold 
him  to  F  at  a  loss  of  50  <fc.     If  F  gave  $150  for  the 
horse,  what  did  D  give  for  him  ? 

6.  At  $7  a  cord,  what  will  a  pile  of  wood,  32  ft. 
long,  4  ft.  wide,  and  6  ft.  high,  cost  ? 

7.  How  much  water  must  be  added  to  21  gal.  of 
alcohol,  worth  $4  a  gallon,  to  make  it  worth  but  $3  a 
gallon  ? 

8.  Four  parties,  having   speculated    with   $6000, 
realized  a  profit  of  20  %.     A  contributed  £  of  the  cap- 
ital, B  ^,  C  -j^-,  and  D  -fa.     The  transaction  being  fin- 
ished, how  much  should  each  get  for  his  share  of  the 
profit  and  capital? 

9.  At  $3.50  a  rod,  what  will  be  the  expense  of 
fencing  a  field  3  rods  square  ? 


154  MISCELLANEOUS    EXAMPLES. 

10.  On  what  day  did  a  note  for  $800  mature,  which 
was  discounted  August  10th,  at  6  %,  for  $2  ? 

11.  What  part  of  a  plot  1  rod  square  is  a  bed  ^  of 
a  rod  square  ? 

12.  How  many  pounds  of  coffee  worth  12c.  a  lb., 
must  be  mixed  with  9  lb.  worth  20c.  a  lb.,  to  make 
the  mixture  worth  15c.  a  lb.  ? 

MODEL.  On  each  pound  put  in  at  20c.,  to  be  sold  at  15c.,  there  is 
a  loss  of  5c. ;  and  on  9  lb.  there  will  be  a  loss  of  9  times  5c.,  or  45c. 
On  each  pound  put  in  at  12c.,  to  be  sold  at  15c.,  there  will  be  a  gain 
of  3c. ;  and,  to  balance  the  loss  of  45c.,  there  must  be  as  many 
pounds  at  12c.  put  in  as  3c.  is  contained  times  in  45c.,  or  15.  Arts. 
15  lb. 

13.  How  many  pounds  of  tea  at  $1.25  a  pound, 
must  be  mixed  with  10  lb.  at  80c.  a  pound,  to  make 
the  mixture  worth  $1  a  pound? 


SECTION  101.— 1.  When  gold  is  quoted  at  150, 
what  is  the  value  in  gold  of  $1  in  currency  ? 

2.  How  many  hours  will  a  person  save  in  the  three 
summer  months  by  sleeping  but  7  hours  daily  in  stead 
of  8? 

3.  How  many  gallons  of  whiskey,  at  $2  a  gallon, 
must  be  mixed  with  12  gal.  worth  $3.50  a  gallon,  to 
make  the  mixture  worth  $3  a  gallon  ? 

4.  Two  clocks  were  sold  for  $60  apiece,  at  a  loss 
of  20  %  on  one,  and  a  profit  of  20  %  on  the  other.     Tak- 
ing both  sales  into  account,  was  there  a  gain  or  loss, 
and  if  either  how  much  ? 

5.  |  of  If  is  f  of  how  many  times  5  %  of  400  ? 


MISCELLANEOUS    EXAMPLES.     %  155 

6.  What  o'clock  is  it,  if  the  time  past  12  is  ^  of  the 
time  from  now  till  1  ? 

V.  How  many  hours  will  you  lose  in  a  leap-year,  if 
you  are  idle  5  minutes  every  day  ? 

8.  In  going  £  of  a  mile,  how  many  more  times  will 
a  wheel  turn  that  is  12  feet  in  circumference,  than  one 
that  is  15  feet? 

9.  If  a  certain  principal,  at  6^,  produces  $135  in  2 
yr.  3  mo.,  how  much  interest  will  the  same  principal 
produce  in  1  yr.  6  mo.,  at  7  %  ? 

10.  The  pipe  A  can  fill  a  cistern  in  |-  of  the  time 
that  the  pipe  B  can  fill  it.     If  both  can  fill  it  in  20 
minutes,  how  many  times  can  the  pipe  A  alone  fill  it 
in  1  hour  ? 

11.  How  many  bushels  of  wheat,  at  $2.50  a  bushel, 
can  be  bought  with  the  proceeds  of  a  note  for  $1500, 
discounted  at  a  bank,  for  2  mo.,  at  6  </0  ? 

12.  Two  oblong  fields  contain  the  same  number  of 
square  rods.     The  first  is  50  rods  long  and  6  rods  wide. 
The  second  is  20  rods  long ;  how  wide  is  it  ? 

13.  Ida  is  now  half  as  old  as  Jane,  but  in  6  years 
she  will  be  f  as  old  as  Jane  will  then  be ;  what  is  the 
age  of  each? 

MODEL.  Ida  is  now  half  as  old  as  Jane.  To  maintain  the  same 
ratio,  she  would  have  to  live  only  3  years  to  Jane's  6 ;  and  there- 
fore at  the  end  of  6  years  she  will  be  half  as  old  as  Jane  will  then 
be,  and  6—3,  or  3,  years  more.  But  by  the  conditions  she  will 
then  be  \  as  old  as  Jane  will  be ;  hence  3  years  must  be  the  differ- 
ence between  ^  and  \  of  Jane's  age  six  years  hence.  Jane  will 
therefore  be  12,  and  Ida  |  of  12,  or  9  ;  and  the  present  age  of  each 
will  be  6  years  less.  Ans.  Jane,  6  ;  Ida,  3. 


156  9    MISCELLANEOUS    EXAMPLES. 

SECTION  102.— 1.  Susan  is  now  3  times  as  old  as 
Sarah  ;  four  years  hence,  her  age  will  be  twice  Sarah's. 
How  old  is  each  ? 

2.  Anna  is  8  years  old.     Ella's  age  equals  Anna's 
increased  by  ^  of  Jacob's ;  and  Jacob's  equals  Ella's 
increased  by  1  £  times  Anna's.     How  old  are  Ella  and 
Jacob  ? 

MODEL.  Anna  being  8  years  old,  Ella's  age  equals  £  of  Jacob's 
+  8  years,  and  Jacob's  equals  Ella's  +  12  years.  Hence  J  of  Jacob's 
age  is  £  of  Ella's  +  4  years  ;  and  Ella's  age  must  equal  £  of  her  age 
+  4  years  +  8  years.  12  years,  therefore,  equals  the  difference  be- 
tween Ella's  age  and  J  of  her  age,  or  is  f  of  Ella's  age.  Hence 
Ella  is  18;  and  Jacob's  age,  being  equal  to  Ella's  increased  by  H 
times  Anna's  age,  or  12  years,  is  SO.  Ans.  Ella,  18  ;  Jacob,  30. 

3.  A  person,  buying  a  horse,  wagon,  and  harness, 
paid  for  the  harness  $40 ;  for  the  wagon  as  much  as 
for  the  harness  and  -J-  of  the  cost  of  the  horse ;  and 
for  the  horse  as  much  as  for  the  wagon  and  twice  the 
cost  of  the  harness.     What  was  the  cost  of  the  whole  ? 

4.  There  are  three  poles,  the  first  of  which  is  as 
long  as  the  other  two  ;  the  second  is  6  feet ;  and  the 
third  is  as  long  as  \  the  first  increased  by  f  of  the 
second.     What  is  the  length  of  the  first  and  third  ? 

5.  How  many  acres  in  an  oblong  garden,  20  rods 
long  by  10  rods  wide  ?     How  many  rods  offence  will 
be  needed  to  enclose  it  ? 

6.  If  |  of  a  piece  of  work  is  done  by  2  men  in  6 
days,  how  many  men  will  it  take  to  do  what  remains 
in  4  days  ? 

7.  What  is   the   amount   of  $600,   for   2  years  3 
months  15  days,  at  6  %  ? 


MISCELLANEOUS    EXAMPLES.  157 

8.  From  what  number  must  f  of  40  be  taken  3 
times,  to  leave  3  ? 

9.  The  product  of  two  numbers  is  36  ;  if  one  of  the 
factors  is  £  of  54,  the  other  is  f  of  what  number? 

10.  Some  apples  were  bought  at  the  rate  of  3  for 
2c.,  and  sold  at  the  rate  of  2  for  3c.     If  the  profit  was 
50c.,  how  many  apples  were  there  ?    "What  was  the 
per  cent,  of  profit  ? 

1!.  A  wall  containing  180  square  feet  is  18  feet 
long;  another  wall,  equally  high,  is  12  feet  long. 
What  is  the  area  of  the  second  wall  ? 

12.  If  4  horses  can  remove  ^  of  a  heap  of  stone  in 
|  of  a  day,  how  many  horses  will  be  needed  to  remove 
the  whole  heap  in  half  a  day  ? 


SECTION  103.— 1.  Divide  $1  between  three  per- 
sons, so  that  the  first  may  have  $T1-U-  more  than  the 
third,  and  $^-  less  than  the  second. 

2.  A  publisher  takes  off  33^  %  from  his  retail  prices 
for  a  wholesale  customer ;  what  f0  will  the  purchaser 
make,  if  he  sells  at  the  publisher's  retail  prices  ? 

3.  A  and  B  are  to  dig  a  cellar  for  $28.     When  J- 
of  the  work  is  done  A  is  taken  sick,  and  B  has  to  fin- 
ish it.     Divide  the  $28  fairly  between  them. 

4.  C  and  D  agreed  to  cut  some  wood  for  $60. 
When  the  work  was  partly  done,  D  was  taken  sick 
and  received  only  $10.    What  part  of  the  job  was  fin- 
ished when  D  quit  work  ? 

5.  f  of  ||  is  8  times  £  of  how  many  times  -£$  ? 


158  MISCELLANEOUS   EXAMPLES. 

6.  What  %  on  the  investment  is  a  2  %  semi-annual 
dividend  on  stock  bought  at  60  %  below  par  ? 

7.  How  many  boxes,  1  ft.  long,  1  ft.  wide,  and  6  in. 
high,  can  be  packed  in  a  space  4  ft.  each  way  ? 

8.  A  lady  bought  some  eggs  at  the  rate  of  3  for  5 
cents,  and  had  lOc.  left.     Had  she  given  25c.  a  dozen, 
she  would  have  needed  5c.  more  to  pay  for  them.     How 
many  did  she  buy  ? 

9.  How  many  yards  of  silk  |  of  a  yd.  wide  will 
line  4  satin  damask  curtains,  each  12  feet  long  by  6 
feet  wide? 

10.  A  person  bought  9  tons  of  coal  for  $88  ;  partly 
soft  coal,  for  which  he  gave  $12  a  ton,  and  partly  hard, 
which  cost  $7.     How  many  tons  of  each  did  he  buy  ? 

11.  Two  blocks  of  stone  contain  the  same  number 
of  cubic  feet.     They  are  both  6  ft.  long  ;  the  first  is  4 
ft,  wide  and  3  ft.  high ;  the  second  is  1  £  ft.  wide, — 
how  high  is  it  ? 

12.  One  number  is  If  times  another  ;  if  the  differ- 
ence between  them  is  10,  what  are  the  numbers  ? 

13.  In  what  time  will  $48.77  amount  to  $97.54,  at 
6£$  ?     At  what  per  cent,  will  it  amount  to  $97.54  in 
12  yr.  6  mo.? 

14.  What  o'clock  is  it,  if  the  time  past  12  is  \  of 
the  time  past  11? 

MODEL.  The  time  past  11  is  }  of  itself;  and,  since  the  time  past 
12  is  4  of  the  time  past  11,  the  time  between  11  and  12,  or  60  min., 
must  be  the  difference  between  \  and  J,  or  jj,  of  the  time  past  11. 
If  60  minutes  are  f  of  the  time  past  11,  -J-  must  be  \  of  60  min.,  or 
30  min.;  and  §  is  3  times  30  minutes,  or  90  minutes.  Ans.  90 
minutes  past  11, — that  is,  half  past  twelve. 


MISCELLANEOUS   EXAMPLES.  159 

SECTION  104. — 1.  A  stage  is  going  at  the  rate  of 
10  miles  in  1J  h. ;  a  train  of  cars  is  coming  up  behind 
at  the  rate  of  10  miles  in  22  min.  If  the  train  over- 
takes the  stage  in  88  minutes,  how  far  was  it  behind 
the  stage  at  first  ? 

2.  What  o'clock  is  it,  if  the  time  past  3  is  ^  of  the 
time  past  1  ? 

3.  What  o'clock  is  it,  if  the  time  past  4  is  ^  of  the 
time  from  now  to  5  ? 

4.  A  crown  is  worth  5s.     How  many  more  books, 
at  half  a  crown  each,  can  be  bought  for  10  guineas 
than  for  £10  ? 

5.  A  person  bought  a  section,  or  square  mile  of 
land,  for  $800.     He  divided  it  into  forty-acre  farms, 
which  he  sold  for  $60  each.     Did  he  gain  or  lose,  and 
what  %  ? 

6.  After  running  at  the  rate  of  20  miles  an  hour 
for  20  minutes,  a  train  increased  its  speed  50  f0 ;  how 
far  did  it  run  in  all  during  the  first  hour  ? 

7.  Three  brothers  have  money  at  interest,  at  6  $, 
from  which  they  realize  $900  yearly.     A  has  f  of  the 
principal,  and  of  the  rest  B  has  4  times  as  much  as  C. 
How  much  has  each? 

8.  An  importer  has  to  pay  a  duty  of  60$  in  gold 
on  silk  invoiced  at  $7000  ;  when  gold  is  at  a  premium 
of  40  $,  what  will  the  duty  cost  in  currency  ? 

9.  A  garrison  of  100  men  have  food  enough  for  60 
days,  allowing  each  man  3  Ib.  a  day.     After  30  days, 
50  more  men  join  them,  and  their  daily  allowance  is 
diminished  half  a  pound.     How  long  will  their  sup- 
plies then  last  ? 


160  EQUATION   OP   PAYMENTS. 

10.  A  owes  B  $400,  payable  in  2  months,  and  $200, 
payable  in  5  months.     At  what  time  should  the  whole 
be  paid,  so  that  neither  party  may  gain  or  lose  ? 

MODEL.  The  use  of  $400  for  2  mo.  is  equivalent  to  the  use  of 
$1  for  4CO  times  2  mo.,  or  800  mo.  The  use  of  $200  for  5  mo.  is 
equivalent  to  the  use  of  $1  for  200  times  5  mo.,  or  1000  mo.  Hence 
A  is  entitled  to  the  use  of  $1  for  800  mo. +  1000  mo.,  or  1800  mo. — 
and  to  the  use  of  the  whole  money  owed  ($400 +  $200,  or  $600)  for 
giu  of  1800  mo.,  or  3  mo.  Ans.  3  months. 

11.  A  merchant  sells  a  customer  $1000  worth  of 
goods  on   3   months'    credit,  and    $500  worth   on   6 
months'.     The  purchaser  wishes  to  make  one  payment 
of  the  whole  ;  when  should  it  be  made? 

12.  A  person  buys  $1800  worth  of  goods;  %  of  the 
bill,  on  1   month's  credit ;  J,  on  2  months' ;  and  the 
rest,  on  4  months'.     In  what  time  may  he  equitably 
make  one  payment  of  the  whole  ? 

13.  When  should  a  party  discharge  a  debt  of  $600 
in  one  payment,  if  75  %  of  it  is  due  in  10  days,  and 
the  rest  in  30  days  ? 

14.  A  trader  bought  a  bill  for  $1000,— half  for  cash, 
and  half  on  3  months'  credit.     If  he  gives  a  note  for 
the  whole,  in  what  time  should  it  mature  ? 

MODEL.  Half  the  bill  being  for  cash,  the  trader  is  entitled  only 
to  the  use  of  $500  for  3  mo. ;  which  is  equivalent  to  the  use  of  $1 
for  1500  mo.,  or  $1000  (the  amount  of  the  bill)  for  7-/0-jj  of  1500  mo., 
or  H  mo.  Ans.  1J  mo. 

15.  When  should  a  bill  for  $4000,  half  cash,  and 
half  due  in  1  mo.,  be  discharged  in  one  payment  ? 

16.  What  is  the  process  treated  of  on  this  page 
called  ?    Ans.  Equation  of  Payments. 


MISCELLANEOUS    EXAMPLES.  161 

SECTION  105.— 1.  A  lent  B  $500  for  6  months ; 
how  long  should  B  lend  A  $1200,  that  neither  may 
gain  or  lose  interest  ? 

2.  Five  times  a  number  exceeds  3  times  16|  </0  of 
the  same  number  by  108  ;  what  is  the  number  ? 

3.  A  person,  being  asked  the  hour,  said  that  the 
time  that  would  elapse  before  midnight  was  f  of  the 
time  past  noon  ;  what  o'clock  was  it  ? 

4.  The  shell  of  a  certain   cocoa-nut  weighs  5  oz. ; 
the  milk  weighs  \  as  much  as  the  shell  added  to  |  the 
weight  of  the  kernel ;  the  kernel  weighs  as  much  as 
the  shell  and  half  the  milk.     What  is  the  weight  of 
the  whole  cocoa-nut  ? 

5.  If  Isaac  had  as  many  more  books  as  he  now  has, 
half  as  many  more,  and  40  volumes  besides,  he  would 
have  twice  five  score  books  ;  how  many  has  he  ? 

6.  A  boy  having  taken  44  steps,  a  man  starts  off 
after  him,  taking  4  steps  to  the  boy's  3.     If  3  of  the 
man's  steps  equal  5  of  the  boy's,  how  many  steps  will 
the  man  take  before  he  comes  up  to  the  boy  ? 

7.  Two  workmen  engage  to  do  a  job  for  $51.     The 
first,  being  the  better  workman,  is  to  have  $9  as  often 
as  the  second  has  $8.     They  finish  the  work  in  8  days  ; 
what  wages  per  day  does  each  make  ? 

8.  If  2  bu.  2  pk.  of  seed  is  allowed  to  an  acre,  how 
much  will  the  seed  for  2  A.  2  B.  cost,  at  $2.50  a  bushel  ? 

9.  How  much  rum,  at  $2  a  gallon,  must  be  mixed 
with  3  gallons  at  $4,  to  make  the  whole  worth  $2.50  a 
gallon  ? 

10.  At  7  ft,,  what  is  the  present  worth  of  80  %  of 
$8025,  due  1  year  hence  without  interest  ? 


162  MISCELLANEOUS    EXAMPLES. 

11.  What  is  P's  income  tax,  if  he  has  an  income  of 
$2400  over  and  above  what  is  exempt,  the  rate  being 
5  £,  and  has  to  pay  on  35  oz.  of  silver  at  5c.  an  oz.  ? 

12.  A  person  insured  his  life  for  83000  at  the  rate 
of  $3.10  on  $100.     After  paying  4  premiums,  he  died; 
how  much  more  did  his  family  receive  than  was  paid 
out  for  premiums  ? 

13.  A  dishonest  milkman  adds  a  quart  of  water  to 
every  gallon  of  milk,  and  then  sells  the  mixture  at  10  $ 
more  a  quart  than  he  paid  for  the  pure  milk.     What  % 
profit  on  the  whole  does  he  make  ? 

14.  A  can  do  a  piece  of  work  in  3  days,  B  in  4 
days,  C  in  5  days.     If  they  do  the  job  together  for 
89.40,  and  each  is  paid  according  to  the  work  he  does, 
how  much  should  each  get  ? 


SECTION  108.— 1.  When  gold  is  130,  is  it  better 
to  lend  money  at  7  $  or  to  buy  five-twenties  at  106  ? 

MODEL.  $100  of  five-twenties,  at  106,  will  cost  $106  in  currency. 
The  yearly  interest  on  $100  of  five-twenties  is  $6  in  gold,  or,  when 
gold  is  130,  $7.80  in  currency.  The  yearly  interest  on  $106  (the 
cost  of  $100  of  five-twenties),  at  7  #,  would  be  §7.42.  Ans.  Five- 
twenties  would  pay  the  better  interest. 

2.  When  gold  is  at  140,  which  is  the  better,  an  in- 
vestment on  bond  and  mortgage  at  6  £,  or  in  five- 
twenties  at  110? 

3.  With  gold  at  150,  is  it  better  to  lend  money  at 
\\%  a  month  or  to  buy  ten-forties  at  105  ? 

4.  Is  it  better  to  buy  State  bonds  paying  6  $,  at 
par,  or  5  $  bonds  at  90  ? 


MISCELLANEOUS   EXAMPLES.  163 

5.  A  pole  is  fixed  in  the  bottom  of  a  river.     Three 
feet  are  in  the  air ;  the  part  in  the  water  is  3  times  as 
long  as  that  in  the  mud ;  and  the  part  in  the  mud  is 
^  of  the  rest  of  the  pole.     How  long  is  the  pole  ? 

6.  Divide  45  into  four  numbers,  each  of  which 
shall  be  half  of  the  next  greater. 

7.  What  fraction  of  a  surface  1  ft.  8  in.  square  is  8 
square  inches  ? 

8.  Two  boys  buy  40  pears  each,  at  2c.  apiece.  One 
sells  his  at  30c.  a  dozen.    The  other  sells  £  of  his  at  cost, 
and  the  rest  at  3c.  apiece.     What  </0  does  each  make  ? 

9.  How  high  must  a  wood-cutter  make  a  pile  of 
wood  which  is  4  ft.  wide  and  36  ft.  long,  to  contain  9 
cords  ? 

10.  Arthur  gave  half  his  marbles  to  Hugh,  who 
gave  £  of  what  he  thus  received  to  Alfred.     Alfred, 
after  winning  6  more,  gave  all  he  had  to  Arthur,  who 
then  found  that  he  had  J  of  his  original  number.  How 
many  had  Arthur  at  first  ? 

11.  Divide  160  into  four  numbers,  each  of  which 
shall  be  3  times  as  great  as  the  next  smaller. 

12.  An  agent  received  $3150,  from  which  he  was 
to  take  his  own  commission  (5  %  on  the  money  invest- 
ed), and  with  the  rest  buy  land,  at  $2  an  acre.     How 
many  acres  did  he  buy  ? 

13.  If  5  men  can  do  as  much  work  as  8  women, 
and  3  women  can  do  as  much  as  5  boys,  and  2  boys 
can  do  as  much  as  3  girls,  how  many  girls  will  it  take 
to  do  as  much  as  4  men  ? 

14.  22  %  of  150  is  what  %  of  the  difference  between 
^  of  200  and  \%  of  500  ? 


164  MISCELLANEOUS   EXAMPLES. 

SECTION  107.— 1.  If  I  ask  for  a  farm  20^  more 
than  it  cost,  but  fall  10  %  on  my  asking  price,  what  % 
do  I  make  by  the  sale  ? 

2.  Bought  5  yd.  of  cloth,  at  $4.50  a  yard ;  2  pair 
of  gloves,  at  $1.25  a  pair ;  3  dress  patterns,  at  85  each ; 
and  \  dozen  handkerchiefs  at  $13  a  dozen.     What  did 
the  bill  amount  to  ? 

3.  A  tenant  has  the  choice  of  paying  $500  rent  in 
advance,  or  $550  at  the  end  of  the  year.     If  he  can 
borrow  the  money  to  make  the  advance  payment  with, 
at  6  %,  which  is  it  better  for  him  to  do  ? 

4.  Three  numbers  multiplied  together  give  a  prod- 
uct of  60.     Two  of  the  factors  are  to  each  other  as  \ 
to  -|,  and  the  third  is  1^.     What  are  the  numbers  ? 

5.  At  $5  a  thousand,  what  is  the  cost  of  24  packs 
of  envelopes,  containing  25  each  ? 

6.  If  a  sovereign  (the  coin  that  represents  £1  ster- 
ling) is  worth  $4.86  in  specie,  what  were  10  sovereigns 
worth  in  currency  when  gold  was  at  150  ? 

7.  From  a  pile  of  wood,  40  ft.  long,  4  ft.  wide,  and 
6  ft.  high,  was  sold  $27  worth,  at  $6  a  cord.     How 
much  was  what  remained  worth,  at  $5|  a  cord  ? 

8.  For  a  cabin  passage  to  Liverpool,  one  steamer 
charges  $120  in  gold,  and  another  $150  in  currency. 
What  premium  must  gold  bring,  to  make  these  prices 
equal ? 

9.  The  interest  on  \  of  A's  fortune  and  -f$  of  B's 
fortune,  for  1  year,  at  7  %,  is  S140.     If  A's  fortune  is  £ 
of  B's,  what  is  the  fortune  of  each  ? 

10.  Mary  is  now  8  yr.  older  than  Ruth  ;  two  years 
ago,  she  was  twice  Ruth's  age.     How  old  is  each  ? 


MISCELLANEOUS    EXAMPLES.  165 

11.  Augustus  e^rns  $15  a  week,  and  determines  to 
save  enough  to  present  his  mother  $50  on  Christmas. 
If  it  is  ten  weeks  to  Christmas,  what  per  cent,  of  his 
salary  must  he  save  ? 

12.  A  dog  overtook  a  fox  after  running  half  a  mile. 
Four  fifths  of  the  distance  the  fox  ran  after  the  dog 
started,  was  8  rods  less  than  6  times  the  start  he  had? 
How  many  rods'  start  had  the  fox  ? 

13.  Fifty  men  have  provisions  to  last  them  60  days, 
at  a  certain  rate  of  supply.     Ten  more  men  coming, 
and  the  daily  supply  being  made  ^  less  than  it  was 
before,  how  long  will  the  provisions  last  ? 

14.  f  of  8  times  f  of  21  is  how  many  times  f  of 


SECTION  108,— 1.  Twelve  bottles,  holding  1  pt. 
3  gi.  each,  being  filled  from  a  cask  of  wine  containing 
30  gallons,  how  much  is  the  rest  of  the  wine  worth,  at 
$1.50  a  quart  ? 

2.  Two  numbers  are  to  each  other  as  5  to  6,  and  \ 
of  their  difference  is  1.     What  are  the  numbers  ? 

3.  If  a  certain  number  increased  by  2  is  multiplied 
by  5,  and  the  product  divided  by  3,  we  shall  have  15. 
What  is  the  number  ? 

NOTE.     What  number  divided  by  3  equals  15  ? 

What  number  multiplied  by  5  equals  45  ? 
What  number  increased  by  2  equals  9  ? 

4.  If  a  certain  number  diminished  by  4  is  divided 
by  %  of  20,  and  the  result  multiplied  by  5,  we  shall 
have  2 £.     What  is  the  number  ? 


166  MISCELLANEOUS    EXAMPLES. 

5.  A  farmer  sold  25  %  of  his  potatoes,  and  then 
25  <f0  of  what  remained.     25  $  of  what  were  then  left 
rotted,  and  he  had  135  bushels  of  good  potatoes  on 
hand.     Ho\v  many  had  he  at  first  ? 

6.  If  you  walk  at  the  rate  of  a  mile  in  20  minutes, 
how  many  miles  can  you  walk  in  the  time  that  you 
would  save  by  sleeping  half  an  hour  less  every  day 
during  the  month  of  July  ? 

7.  What  will  it  cost  to  insure  a  house  worth  $6000, 
and  furniture  in  it  worth  $3000,  for  two  thirds  of  their 
value,  at  -^  of  1  $,  5  %  being  deducted  from  the  pre- 
mium for  cash  ? 

8.  A  bankrupt  failed,  having  $7500  liabilities,  and 
$1500  assets.     How  much  could  he  pay  on  the  dollar? 
What  was  B's  loss,  whom  he  owed  for  12  pieces  of 
muslin,  containing  40  yd.  each,  at  25c.  a  yd.  ? 

9.  What  cost  2  bu.  3  pk.  6  qt.  of  oats,  at  75c.  a  bu.  ? 

10.  If  a  grocer  in  10  days  sells  12  cwt.  20  Ib.  of 
sugar,  what  do  his  daily  sales  of  sugar  bring  in,  on  an 
average,  at  12c.  a  pound  ? 

11.  If  2  hands  are  f  of  a  span,  if  a  span  is  |  of  a 
cubit,  and  a  cubit  £  of  a  fathom,  how  many  feet  are 
10  fathoms  equal  to,  the  hand  being  4  inches  ? 

12.  How  many  cubes  \  ft.  long,  wide,  and  high, 
can  be  packed  in  a  space  1  ft.  long,  wide,  and  high  ? 

13.  If  there  are  3  miles  in  a  league,  and  a  boat  has 
to  go  10  leagues,  what  part  of  the  distance  has  she 
made  when  she  has  gone  160  rods  ? 

14.  From  2  qt.  of  seed  was  raised  1£  bu.  of  grain ; 
what  per  cent,  of  the  seed  was  the  crop  ? 

15.  £  of  18  is  how  many  tenths  of  \  of  60  ? 


MISCELLANEOUS    EXAMPLES.  167 

SECTION  109.— 1.  At  what  time  between  12  and 
1  will  the  minute  and  hour  hand  of  a  clock  point  in 
exactly  opposite  directions  ? 

MODEL.  The  minute  and  hour  hand  point  in  exactly  opposite  di- 
rections at  6  o'clock.  Within  the  next  12  hours  they  will  point  in 
exactly  opposite  directions  11  times,  each  time  requiring  -j^-of  12  h., 
or  -^  h.,  to  get  in  this  position.  When  they  stand  in  opposite  di- 
rections between  12  and  1,  it  is  the  sixth  time  they  have  so  stood 
since  6  o'clock,  and  it  is  therefore  6  times  jf  h.,  or  }^  h.,  after  6 ; 
which  makes  the  time  32-ft-  min.  past  12.  Ans.  3 2 -ft-  min.  past  12. 

2.  At  what  time  between  7  and  8  will  the  hour  and 
minute  hand  of  a  watch  point  in  exactly  opposite  di- 
rections ?    At  what  time  between  1  and  2  ? 

3.  A  pasture  is  hired  for  $23.     B  puts  in  twice  as 
many  cows  as  C,  and  C  3  times  as  many  as  D.     D's 
cows  are  in  twice  as  long  as  C's,  and  B's  3  times  as 
long  as  C's.     How  much  should  each  pay  ? 

4.  G  and  H  embarked  in  a  speculation.     G  fur- 
nished $400  for  2  months,  and  received  $120  for  his 
share  of  the  profit.     H  furnished  $600  for  1  month, 
and  received  a  proportionate  share  of  the  profit.   What 
</0  of  the  whole  capital  was  the  profit  ? 

5.  A  could  dig  a  certain  cellar  in  12  days,  B  in  10 
days,  and  C  in  15  days.     They  all  worked  one  day, 
when  A  quit.     B  and  C  worked  the  next  day,  when  B 
quit.     How  long  did  it  take  C  to  finish  it  ? 

6.  Ira's  age  is  6,  Paul's  20 ;  in  how  many  years 
will  Paul  be  twice  as  old  as  Ira  ? 

MODEL.  Paul  is  14  years  older  than  Ira ;  hence,  when  Ira  is  14, 
Paul  will  be  28,  or  twice  as  old  as  Ira.  But,  as  Ira  is  now  6,  he  will 
be  14  in  8  years.  Am.  8  yr. 


168  MISCELLANEOUS    EXAMPLES. 

7.  In  how  many  years  will  Stephen's  age  be  half 
of  Andrew's,  if  Stephen  is  3  and  Andrew  12  ? 

8.  Two  horses  trot  in  the  same  direction  round  a 
circular  course  1  mile  long.     One  goes  at  the  rate  of 
6  miles  an  hour,  the  other  10.     How  many  minutes 
after  starting  will  they  be  together  again  ? 

9.  Two   brothers    having    different    amounts    of 
money,  the  elder  equalized  them  by  giving  the  younger 
as  much  as  he  already  had.     If  the  elder  originally 
had  $6000,  how  much  had  the  younger  ? 

NOTE.  After  the  gift  each  brother  had  twice  as  much  as  the 
younger  had  at  first ;  and,  as  the  elder  had  given  away  as  much  as 
the  younger  had  at  first,  he  must  have  had  at  first  3  times  as  much 
as  the  younger. 

10.  Having  60  ducks  in  one  coop  and  a  smaller 
number  in  another,  I  put  with  the  latter  twice  their 
own  number  from  the  other  coop,  and  then  found  the 
numbers  in  the  coops  equal ;  how  many  ducks  had  I  ? 

11.  A  man  sold  whiskey  that  cost  $2.50  a  gall,  for 
$3.     The  price  of  this  whiskey  having  advanced  to 
$3,  he  watered  it  so  that  he  could  still  sell  it  at  $3  and 
make  the  same  per  cent,  as  before.     How  much  water 
did  he  put  with  10  gal.  of  whiskey  ? 

12.  A,  having  ^  of  a  mile  the  start  of  B,  in  the 
next  half  mile  he  runs  gains  on  B  10  rd.  more ;  after 
which  B  runs  3  rd.  to  A's  2.     How  far  in  all  did  each 
run  before  A  was  overtaken  ? 

13.  There  are  3  boxes,  the  first  of  which  weighs 
10  lb.,  which  is  -^  of  the  weight  of  the  -second  and 
third,  while  the  third  weighs  f  as  ranch 

two.     What  is  the  weight  of  all  t\ 

UNIVERSITY  ) 


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SPANISH. 

A  NEW  METHOD  OF  LEARNING  TO  READ,  WRITE, 
and  Speak  the  Spanish  Language,  after  the  System  of  Ollendorff. 
By  Mno.  Velazquez  and  T.  Simonne.  1  vol.,  12mo,  560  pages. 
Price,  $1.30. 

KEY  TO   THE   EXERCISES  IN   THE  NEW  METHOD  OF 

Learning  to  Read,  Write,  and  Speak  the  Spanish  Language, 
after  the  System  of  Cllendorff.  By  M.  Velazquez  and  T.  Si- 
monne.  1  rol.,  12mo.  174  pages.  Price,  85  cents. 

NUEVO  METODO  PARA  APRENDER  A  LEER,  HABLAR, 
y  Escribir  el  Espafiol,  segun  el  Sistema  dc  Ollendorff.  Para  uso 
de  los  Alemane*.  Arreglado  por  D.  H.  Wrage  y  H.  M.  Mon- 
santo. (Neue  Methode  die  Spanische  Sprache  lesen,  eprechen 
und  schreiben  zu  lernen,  nach  dem  Ollendorff'schen  System.) 
1  TO!.,  12mo.  Price,  $1.50. 

CLAVE  DEL  ANTERIOR.     Price,  $1.00. 

METODO  PARA  APRENDER  A  LEER,  ESCRIBIR  Y  HA- 
blar  el  Ingles,  segun  el  Sistema  de  Ollendorff.  Por  Ramon 
Palenzuela  y  Juan  de  la  C.  Carrefio.  Un  tomo  de  457  paginas, 
en  12°.  Price,  $1.50. 

CLAVE  DE  LOS  EJERCICIOS  DEL  METODO  PARA 
aprender  a  Leer,  Escribir  y  Hablar  el  Ingles,  segun  el  Sisteraa 
de  Ollendorff.  Por  Ramon  Palenzuela  y  Juan  de  la  C.  Carreno. 
Un  tomo  de  111  pagiuas,  en  12°.  Price,  $1.00. 

UN  METODO  PARA  APRENDER  A  LEER,  ESCRIBIR 
y  IJablar  el  Frances  segun  el  Sistema  de  Ollendorff.  Por  Teodoro 
Simonne.  Un  toino  de  341  paginas,  en  12Q.  Price,  $1.50. 

CLAVE  DE  LOS  EJERCICIOS  DEL  METODO  PARA 
aprender  a  Leer,  Escribir  y  Hablar  el  Frances  segun  el  Sistem* 
de  Ollendorff.  Por  Teodoro  Simonne.  Un  tomo  de  80  paginas, 
en  12°.  Price,  $1.00. 


D.  APPLETON  &  CO ,  549  &  551  Broadway,  New  York. 


PRIMERS 

IN  SCIENCE,  HISTORY,  AND  LITERATURE. 

18mo.      .      .     Flexible  cloth,  45  cents  each. 


I.—  Edited    by   Professors    HUXLEY,   ROSCOE,    and    BALFOUR 
STEWART. 


Chemistry H.  E.  ROSCOE. 

Physics BALFOUR  STEWART. 

Physical  Geography,  A.  GEI- 

KIE. 

Geology A.  GEIKIK. 

Physiology M.  FOSTER. 


Astronomy  ......  J.  N.  LOCKYER. 

Botany  ...........  J.  D.  HOOKER. 

Logic  .............  W.  S.  JEVONS. 

Inventional  Geometry,  W.  G. 

SPENCER. 
Pianoforte.  ..FRANKLIN  TAYLOR. 


Political  Economy W.  S.  JEVONS. 

II.— Edited  by  J.  R.  GREEN.  M.A., 
Examiner  in  the  School  of  Modern  History  at  Oxford. 


Greece C.  A.  FYFFE. 

Koine M.  CREIGHTON. 

Europe E.  A.  FREEMAN. 


Old  Greek  Life,  J.  P.  MAHAFFY. 
Koman  Antiquities,  A.  S.  WIL- 

KINS. 

Geography GEORGE  GROVE. 


History  of  Europe E.  A.  FREEMAN. 

III.— Edited  by  J.  R.  GREEN,  M.  A. 


English  Grammar.. R.  MORRIS. 
English  Literature,  STOPFORD 
BROOKE. 

Philology J.  PEILE. 

Classical    Geography,    M.    F. 


TOZER. 


Shakespeare  ........  E.  DOTVDEN. 

Studies  in  Bryant....  J.  ALDEN. 

Greek  Literature...  R.  C.  JEBB. 
English  Grammar  Exercises, 

R.  MORRIS. 
Homer  .......  W.  E.  GLADSTONE. 


(Others  in  preparation.) 


The  object  of  these  primers  is  to  convey  information  in  such  a 
manner  as  to  make  it  both  intelligible  and  interesting  to  very  younir 
papils,  and  so  to  discipline  their  minds  as  to  incline  them  to  more 
systematic  after-studies.  They  are  not  only  an  aid  to  the  pupil,  but 
to  the  teacher,  lightening  the  task  of  each  by  an  agreeable,  easy,  and 
natural  method  of  instruction.  In  the  Science  Series  some  simple 
experiments  have  been  devised,  leading  up  to  the  chief  truths  of  each 
science.  By  this  means  the  pupil's  interest  i*  excited,  and  the  memory 
is  impressed  so  as  to  retain  without  difficulty  the  facts  brought  under 
observation.  The  woodcuts  which  illustrate  these  primers  serve  the 
same  purpose,  embellishing  and  explaining  the  text  at  the  same  time. 

D.  APPLETON  &  CO.,  Publishers,  Nnv  York. 


JUST    PUBLISHED. 


COXSISTIXG    OF  FIVE  BOOKS. 

By  W>r.  T.  HARRIS,  LL.  D.,  Supt.  of  Schools,  St.  Louis,  Mo.;  ANDREW 
J.  EICKOFF,  A.  M.,  Supt.  of  Instruction,  Cleveland,  O. ;  and 
MARK  BAILEY,  A.M.,  Instructor  in  Elocution,  Yale»College. 

Appletons' First  Reader 90  pages. 

Appletons'  Second  Header 142     " 

Appletons'  Third  Reader 214     " 

Appletons'  Fourth  Reader 248     " 

Appletons'  Fifth  Reader 460     " 


These  Readers,  while  avoiding  extremes  and  one-sided  ten- 
dencies, combine  into  one  harmonious  whole  the  several  results 
desirable  to  be  attained  in  a  series  of  school  reading-books. 
These  include  good  pictorial  illustrations,  a  combination  of  the 
word  and  phonic  methods,  careful  grading,  drill  on  the  peculiar 
combinations  of  letters  that  represent  vowel-sounds,  correct 
spelling,  exercises  well  arranged  for  the  pupil's  preparation  by 
himself  (so  that  he  shall  learn  the  great  lessons  of  self-help, 
self-dependence,  the  habit  of  application),  exercises  that  develop 
a  practical  command  of  correct  forms  of  expression,  good  lit- 
erary taste,  close  critical  power  of  thought,  and  ability  to  in- 
terpret the  entire  meaning  of  the  language  of  others. 

The  high  rank  which  the  authors  have  attained  in  the  edu- 
cational field  and  their  long  and  successful  experience  in  prac- 
tical school-work  especially  fit  them  for  the  preparation  of 
text-books  that  embody  all  the  best  elements  of  modern  educa- 
tive ideas.  In  the  schools  of  St.  Louis  and  Cleveland,  over 
which  two  of  them  have  long  presided,  the  subject  of  reading 
lias  received  more  than  usual  attention,  and  with  results  that 
have  established  for  them  a  wide  reputation  for  superior  elo- 
cutionary discipline  and  accomplishments. 

Of  Prof.  Bailey,  Instructor  of  Elocution  in  Yale  College,  it 
is  needless  to  speak,  for  he  is  known  throughout  the  Union  as 
being  without  a  peer  in  his  profession.  His  methods  make  nat- 
ural, not  mechanical  readers. 


D,  APPLETON  &  CO.,  Publishers,  549  &  551  Broadway,  New  York. 


YA  02420 


/ 


ARITHMETICAL  *SERIES, 

By  G-.  P.  QUACKENBOS,  LL.  D. 
Cpen  the  tais  of  the  tt  erk8  tf  GEORGE  B.  PEBFJKS,  IX.  D. 

I.  A  PRIMARY  ARITHMETIC.—  Beautifully  Illustrated ;  carries 
the  beginner  through  the  first  four  Rules  »uJ  the  simple  Tables,  combining 
mental  examples  with  sum*  for  the  slate.  16mo,  108  pages.  22  cents. 

H.  AH  ELEMENTARY  ARITHMETIC.— Reviews  the  sub- 
jects  of  the  Primary  in  a  style  adapted  to  somewhat  maturer  minds;  also  em- 
braces Fractions.  Fedoral  Money,  Reduction,  and  the  Compound  Rule*.  12mo, 
144  pages.  40  cents. 

m.  A  PRACTICAL  ARITHMETIC.— FulU  clear,  condensed. 
Prepared  with  nco  to  the  wants  of  Common  Schools,  and  designed 

to  prepare-  pupils  thoro^hly  for  the  business  of  life.  Its  methods  are  those 
actually  used  by  busies  men.  12mo,  336  pages.  80  cents. 


IV.  A  HIGKffER  ARITHMETIC.— This  work  carries  the  subject 
further,  an*  «o.  •  thorough  mastery  of  tb«  theory 
and  pra<  It  is  really  whr.i  its  name  imports 

|  PaS 

V.  A'ME?-*.L/*L  ARITHBtBTIC.— For    imparting  read  ; 
mental  ealcuh.  new  and  beautiful  processes.    16mo,  j 
35  cent*. 


The  ditf  mber?   of    this  Series  are  perfectly1! 

graded.  i'rom  step   to  step  is  inductive  a 

gradual ;  i  anticipated,    nothing  required  to  be 

supplied  by  the  teacher;      The  definitions  are  simple,  the 
rules  brief,  the  analyses  unencumbered  with  verbiage,  the 
examples  drawn  from  the    practical  matters  of  life,  the 
arrangement   the  most  natural,  the  methods  taught  the: 
shortest  possible.      Every  device  is  resorted  to,  to  prevent  i 
the  mere  mechanical    solving  of   examples ;  the^  pupil's  I 
mind  is  constantly  kept  on  the  alert,  and  his  Arithmetic 
letson  is  thus  made  an  invaluable  mental  discipline. 


T»aohert  that  want  the  best  books  should  examine  the  abore, 


